perm filename V2S.IN[1,DEK] blob sn#285518 filedate 1977-05-29 generic text, type C, neo UTF8
COMMENT ⊗   VALID 00014 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	folio 802 galley 1
C00020 00003	folio 805 galley 2
C00030 00004	folio 810 galley 3
C00053 00005	folio 818 galley 4
C00075 00006	folio 821 galley 5
C00096 00007	folio 824 galley 6
C00116 00008	folio 828 galley 7
C00137 00009	folio 832 galley 8
C00160 00010	folio 835 galley 9
C00182 00011	folio 842 galley 10
C00197 00012	folio 844 galley 11 "special italic figs in position 4"
C00213 00013	folio 845 galley 12
C00230 00014	folio 849 galley 13
C00242 ENDMK
C⊗;
folio 802 galley 1
    0  {U0}{H9L11M29}|πW58320#Computer Programming!(Knuth/Addison-W
    1  esley)!Answers!f.|4802!g.|41d|'{A6}!!|1|1This 
    3  extension of Euclid's algorithm includes most 
    9  of the features we have seen in previous extensions, 
   18  all at the same time, so it provides new insight 
   28  into the special cases already considered. To 
   35  prove that it is valid, note _rst that deg(|εv|β2) 
   44  |πdecreases in step C4, so the algorithm certainly 
   52  terminates. At the conclusion of the algorithm, 
   59  |εv|β1 |πis a common right divisor of |εV|β1 
   67  |πand |εV|β2, |πsince |εw|β1v|β1|4α=↓|4(|→α_↓1)|gnV|β1 
   71  |πand |ε|→α_↓w|β2v|β1|4α=↓|4(|→α_↓1)|gnV|β2; 
   73  |πalso if |εd |πis any common right divisor of 
   82  |εV|β1 |πand |εV|β2, |πit is a right divisor 
   90  of |εz|β1V|β1|4α+↓|4z|β2V|β2|4α=↓|4v|β1. |πHence 
   93  |εv|β1|4α=↓|4|πgerd(|εV|β1,|4V|β2). |πAlso if 
   96  |εm |πis any common left multiple of |εV|β1 |πand 
  105  |εV|β2, |πwe may assume without loss of generality 
  113  that |εm|4α=↓|4U|β1V|β1|4α=↓|4U|β2V|β2, |πsince 
  116  the sequence of values of |εQ |πdoes not depend 
  125  on |εU|β1 |πand |εU|β2. |πHence |εm|4α=↓|4(|→α_↓1)|gn(|→α_↓u
  130  |β2z|ur|↔0|)1|))V|β1|4α=↓|4(|→α_↓1)|gn(u|β2z|ur|↔0|)2|))V|β2
  130   |πis a multiple of |εz|ur|↔0|)1|)V|β1.|'!!|1|1|πIn 
  137  practice, if we just want to calculate gerd(|εV|β1,|4V|β2), 
  145  |πwe may suppress the computation of |εn, w|β1, 
  153  w|β2, w|ur|↔0|)1|), w|ur|↔0|)2|), z|β1, z|β2, 
  158  z|ur|↔0|)1|), z|ur|↔0|)2|); |πThese additional 
  162  quantities were added to the algorithm primarily 
  169  to make its validity more readily established.|'
  176  !!|1|1|ε|*/Note:|\ |πNontrivial factorizations 
  179  of string polynomials, such as the example given 
  187  with this exercise, can be found from matrix 
  195  identities such as|'{A9}|↔a|(|εa|1!!|d51!!0|)|↔s|↔a|(b!!1|d5
  198  1!!0|)|↔s|↔a|(|1c|1!!1|d51!!0|)|↔s|↔a|(0!!1|d51!|5|→α_↓|1c|1
  198  |)|↔s|↔a|(0!!1|d51!|5|→α_↓b|)|↔s|↔a|(0!!1|d51!|5|→α_↓a|1|)|↔
  198  s|4α=↓|4|↔a|(1!!0|d50!!1|)|↔s,|;{A9}|πwhich hold 
  201  even when multiplication is not commutative. 
  207  For example,|'{A9}|ε(abc|4α+↓|4a|4α+↓|4c)(1|4α+↓|4ba)|4α=↓|4
  209  (ab|4α+↓|41)(cba|4α+↓|4a|4α+↓|4c).|;{A9}(|π(Compare 
  211  this with the ``continuant polynomials'' of Section 
  218  4.5.3.)|'{A3}|≡1|≡9|≡.|9|4(Solution by Michael 
  222  Fredman.) If such an algorithm exists, |εD |πis 
  230  a gerd by the argument in exercise 18. Let us 
  240  regard |εA |πand |εB |πas a single |ε2n|4α⊗↓|4n 
  248  |πmatrix |εC |πwhose _rst |εn |πrows are those 
  256  of |εA, |πand whose second |εn |πrows are those 
  265  of |εB. |πSimilarly, |εP |πand |εQ |πcan be combined 
  274  into a |ε2n|4α⊗↓|4n |πmatrix |εR; X |πand |εY 
  282  |πcan be combined into an |εn|4α⊗↓|42n |πmatrix 
  289  |εZ: |πthe desired conditions now reduce to two 
  297  equations |εC|4α=↓|4RD, D|4α=↓|4ZC. |πIf we can 
  303  _nd a |ε2n|4α⊗↓|42n |πinteger matrix |εU |πof 
  310  determinant |→|¬N1 such that the last |εn |πrows 
  318  of |εU|gα_↓|g1C |πare all zero, then |εR|4α=↓|4(|π_rst 
  325  |εn |πcolumns of |εU), D|4α=↓|4(|π_rst |εn |πrows 
  332  of |εU|gα_↓|g1C), Z|4α=↓|4(|π_rst |εn |πrows 
  337  of |εU|gα_↓|g1) |πsolves the desired conditions. 
  343  Hence, for example, the following triangularization 
  349  algorithm may be used (|εm|4α=↓|42n):|'{A3}!!|1|1|π|≡A|≡l|≡g
  354  |≡o|≡r|≡i|≡t|≡h|≡m |≡T|≡.|9|4Let |εC |πbe an 
  359  |εm|4α⊗↓|4n |πmatrix of integers. This algorithm 
  365  _nds |εm|4α⊗↓|4m |πinteger matrices |εU |πand 
  371  |εV |πsuch that |εUV|4α=↓|4I |πand |εVC |πis 
  378  ``upper triangular.'' (The entry in row |εi |πand 
  386  column |εj |πof |εVC |πis zero if |εi|4|¬Q|4j.)|'
  394  {A3}{I3.1H}|π!!|1|1|≡T|≡1|≡.|9Set |εU|4|¬L|4V|4|¬L|4I, 
  396  |πthe |εm|4α⊗↓|4m |πidentity matrix; and set 
  402  |εT|4|¬L|4C. (|πThroughout the algorithm we will 
  408  have |εT|4α=↓|4VC, UV|4α=↓|41.)|'{A3}!!|1|1|π|≡T|≡2|≡.|9Do 
  412  step T3 for |εj|4α=↓|41,|42,|4.|4.|4.|4,|4|πmin(|εm,|4n), 
  416  |πand terminate the algorithm.|'{A3}!!|1|1|≡T|≡3|≡.|9Perform
  420   the following transformation zero or more times 
  428  until |εT|βi|βj |πis zero for all |εi|4|¬Q|4j: 
  435  |πLet |εT|βk|βj |πbe a nonzero element of |¬T|εT|βi|βj,|4T|β
  442  (|βj|βα+↓|β1|β)|βj,|4.|4.|4.|4,|4T|βm|βj|¬Y |πhaving 
  444  the smallest absolute value. Interchange rows 
  450  |εk |πand |εj |πof |εT |πand of |εV; |πinterchange 
  459  columns |εk |πand |εj |πof |εU. |πThen subtract 
  467  |"l|εT|βi|βj/T|βj|βj|"L |πtimes row |εj |πfrom 
  472  row |εi, |πin matrices |εT |πand |εV, |πand add 
  481  the same multiple of column |εi |πto column |εj 
  490  |πin matrix |εU, |πfor |εj|4|¬W|4i|4|¬E|4m.|'
  495  {A3}{IC}!!|1|1|πFor the stated example, the algorithm 
  501  yields (|f1!!2|d53!!4|))|4α=↓|4(|f1!!0|d53!!2|))(|f1!!2|d50!
  502  |5|→α_↓1|)), (|f4!!3|d52!!1|))|4α=↓|4(|f4!!5|d52!!3|))(|f1!!
  503  2|d50!|5|→α_↓1|)), (|f1!!2|d50|5!|→α_↓1|))|4α=↓|4(|f1!!0|d52
  504  !|5|→α_↓2|))(|f1!!2|d53!!4|))|4α+↓|4(|f0!!0|d51!!0|))(|f4!!3
  504  |d52!!1|)). (Actually any matrix with determinant 
  510  |→|¬N1 would be a gcrd in this case.)|'{A3}|≡2|≡0|≡.|9|4It 
  519  may be helpful to consider exercise 4.6.2<22 
  526  with |εp|gm |πreplaced by a small number |ε|≤e.|'
  534  {A3}|≡2|≡1|≡.|9|4|πNote that Algorithm R is used 
  540  only when |εm|4α_↓|4n|4|¬E|41, |πand that the 
  546  coe∃cients are bounded by (25) with |εm|4α=↓|4n. 
  553  [|πThe stated formula is, in fact, the execution 
  561  time observed in practice, not merely an upper 
  569  bound. For more detailed information see G. E. 
  577  Collins, |εProc. |*/|↔O|↔m|↔o|↔l |\Summer Inst. 
  582  On Symbolic Math. Comp., |πRobert G. Tobey, ed. 
  590  (IBM Federal Systems Center, June 1969), 195<231.]|'
  597  {A3}|≡2|≡2|≡.|9|4A sequence of signs cannot contain 
  603  two consecutive zeros, since |εu|βk|βα+↓|β1(x) 
  608  |πis a nonzero constant in (28). Moreover we 
  616  cannot have |→α+↓,|40,|4|→α+↓ or |→α_↓,|40,|4|→α_↓ 
  621  as subsequences. The formula |εV(u,|4a)|4α_↓|4V(u,|4b) 
  626  |πis clearly valid when |εb|4α=↓|4a, |πso we 
  633  must only verify it as |εb |πincreases. The polynomials 
  642  |εu|βj(x) |πhave _nitely many roots, and |εV(u,|4b) 
  649  |πchanges only when |εb |πencounters or passes 
  656  such roots. Let |εx |πbe a root of some (possibly 
  666  several) |εu|βj. |πWhen |εb |πincreases from 
  672  |εx|4α_↓|4|≤e |πto |εx, |πthe sign sequence near 
  679  |εj |πgoes from |→α+↓,|4|→|¬N,|4|→α_↓ to |→α+↓,|40,|4|→α_↓ 
  685  or from |→α_↓,|4|→|¬N,|4|→α+↓ to |→α_↓,|40,|4|→α+↓ 
  690  if |εj|4|¬Q|40; |πand from |→α+↓,|4|→α_↓ to 0,|4|→α_↓ 
  697  or from |→α_↓,|4|→α+↓ to 0,|4|→α+↓ if |εj|4α=↓|40. 
  704  (|πSince |εu|¬S(x) |πis the derivative, |εu|¬S(x) 
  710  |πis negative when |εu(x) |πis decreasing.) Thus 
  717  the net change in |εV |πis |→α_↓|ε|≤d|βj|β0. 
  724  |πWhen |εb |πincreases from |εx |πto |εx|4α+↓|4|≤e, 
  731  |πa similar argument shows that |εV |πremains 
  738  unchanged.|'!!|1|1[L. E. Heindel, |εJACM |π|≡1|≡8 
  744  (1971), 533<548, has applied these ideas to construct 
  752  algorithms for isolating the real zeroes of a 
  760  given polynomial |εu(x), |πin time bounded by 
  767  a polynomial in deg(|εu) |πand log|4|εN, |πwhere 
  774  all coe∃cients |εy|βj |πare integers with |¬G|εu|βj|¬G|4|¬E|
  780  4N, |πand all operations are guaranteed to be 
  788  exact.]|'{A3}|≡2|≡3|≡.|9|4If |εv |πhas |εn|4α_↓|41 
  793  |πreal roots occurring between the |εn |πreal 
  800  roots of |εu, |πthen (by considering sign changes) 
  808  |εu(x)|πmod|4|εv(x) |πhas |εn|4α_↓|42 |πreal 
  812  roots lying between the |εn|4α_↓|41 |πof |εv.|'
  819  {A24}{H10}|π|∨S|∨E|∨C|∨T|∨I|∨O|∨N |∨4|∨.|∨6|∨.|∨2|'
  821  {A12}{H9}|9|1|≡1|≡.|9|4|πBy the principle of 
  825  inclusion and exclusion (Section 1.3.3), the 
  831  number of polynomials without linear factors 
  837  is |¬K|ε|βk|β|¬E|βn|4(|fp|d5k)p|gn|gα_↓|gk(|→α_↓1)|gk|4α=↓|4
  838  p|gn|gα_↓|gp(p|4α_↓|41)|gp. |πThe stated probability 
  842  is therefore 1|4α_↓|4(1|4α_↓|41/|εp)|gp, |πwhich 
  846  is greater than |f1|d32|); in fact, it is greater 
  855  than |ε|f1|d32|) |πfor all |εn|4|¬R|41.|'{A3}|π|9|1|≡2|≡.|9|
  860  4(a) We know that |εu(x) |πhas a representation 
  868  as a product of irreducible polynomials, and 
  875  the leading coe∃cients of these polynomials must 
  882  be units, since they divide the leading coe∃cient 
  890  of |εu(x). |πTherefore we may assume that |εu(x) 
  898  |πhas a representation as a product of monic 
  906  irreducible polynomials |εp|β1(x)|ge|r1|4.|4.|4.|4p|βr(x)|ge
  908  |rr, |πwhere |εp|β1(x),|4.|4.|4.|4,|4p|βr(x) 
  911  |πare distinct. This representation is unique, 
  917  except for the order of the factors, so the conditions 
  927  on |εu(x), v(x), w(x) |πare satis_ed if and only 
  936  if|'{A9}|εv(x)|4α=↓|4p|β1(x)|g|"l|ge|r1|g/|g2|g|"L|4.|4.|4.|
  937  4p|βr(x)|g|"l|ge|rr|g/|g2|"L, w(x)|4α=↓|4p|β1(x)|ge|r1|1|1|π
  938  |gm|go|gd|1|1|g2|4.|4.|4.|4|εp|βr(x)|ge|rr|1|1|π|gm|go|gd|1|
  938  1|g2.|;{A9}!!|1|1(b) The generating function 
  943  for the number of monic polynomials of degree 
  951  |εn |πis |ε1|4α+↓|4pz|4α+↓|4p|g2z|g2|4α+↓|4αo↓|4αo↓|4αo↓|4α=
  953  ↓|41/(1|4α_↓|4pz). |πThe generating function 
  957  for the number of polynomials of degree |εn |πhaving 
  966  the form |εv(x)|g2, |πwhere |εv(x) |πis monic, 
  973  is 1|4α+↓|4|εpz|g2|4α+↓|4p|g2z|g4|4α+↓|4αo↓|4αo↓|4αo↓|4α=↓|4
  974  1/(1|4α_↓|4pz|g2). |πIf the generating function 
  979  for the number of monic squarefree polynomials 
  986  of degree |εn |πis |εg(z), |πthen by part (a) 
  995  1/(1|4α_↓|4|εpz)|4α=↓|4g(z)/(1|4α_↓|4pz|g2). 
  996  |πHence |εg(z)|4α=↓|4(1|4α_↓|4pz|g2)/(1|4α_↓|4pz)|4α=↓|41|4α
  997  +↓|4pz|4α+↓|4(p|g2|4α_↓|4p)z|g2|4α+↓|4(p|g3|4α_↓|4p|g2)z|g3|
  997  4α+↓|4αo↓|4αo↓|4αo↓|4. |π|πThe answer is |εp|gn|4α_↓|4p|gn|g
 1001  α_↓|g1 |πfor |εn|4|¬R|42. (|πCuriously, this 
 1006  proves that gcd{H11}({H9}|εu(x),|4u|¬S(x){H11}){H9}|4α=↓|4|π
 1008  1 with probability 1|4α_↓|41/|εp; |πit is the 
 1015  sameas the probability that gcd({H11}({H9}|εu(x),|4v(x){H11}
 1019  ){H9}|4α=↓|41 |πif |εu(x) |πand |εv(x) |πare 
 1025  |εindependent, |πby exercise 4.6.1<5.)|'{A3}|9|1|≡3|≡.|9|4Le
 1029  t |εu(x)|4α=↓|4u|β1(x)|4.|4.|4.|4u|βr(x). |πThere 
 1032  is |εat most |πone such |εv(x), |πby the argument 
 1041  of Theorem 4.3.2C. |πThere is |εat least |πone 
 1049  if, for each |εj, |πwe can solve the system with 
 1059  |εw|βj(x)|4α=↓|41 |πand |εw|βk(x)|4α=↓|40 |πfor 
 1063  |εk|4|=|↔6α=↓|4j. |πA solution to the latter 
 1069  is |εv|β1(x)|4|≥7|βk|=|β|↔6|βα=↓|βj|4u|βk(x), 
 1071  |πwhere |εv|β1(x) |πand |εv|β2(x) |πcan be found 
 1078  satisfying|'{A9}|εv|β1(x)|4|≥7|βk|=|β|↔6|βα=↓|βj|4u|βk(x)|4α
 1079  +↓|4v|β2(x)u|βj(x)|4α=↓|41,!!|πdeg(|εv|β1)|4|¬W|4|πdeg(|εu|β
 1079  j),|;{A9}|πby the extension of Euclid's algorithm 
 1086  (exercise 4.6.1<3).|'{A1}|Ha{U31}TEAM|91|'{H9L11M29}W58320#|
folio 805 galley 2
 1089  πComputer Programming|9(Knuth/Addison-Wesley)|9Answers|9f.|4
 1090  805|9g.|42d|'{A6}|5|5|≡4|≡.|9|4|πThe unique factorization 
 1094  theorem gives the identity (|ε1|4α_↓|4pz)|gα_↓|g1|4α=↓|4|≥7|
 1098  ur|)n|¬R1|)|4(1|4α_↓|4z|gn)|gα_↓|ga|rn|rp; |πafter 
 1100  taking logarithms, this can be rewritten |¬K|ur|)j|¬R1|)|4G|
 1106  βp(z|gj)/j|4α=↓|4|¬K|ur|)k,j|¬R1|)|4a|βk|βpz|gk|gj/j|4α=↓|4|
 1106  πln{H11}({H9}1/(1|4α_↓|4pz){H11}){H9}. |πThe 
 1108  stated identity now yields the answer |εG|βp(z)|4α=↓|4|¬K|ur
 1114  |)m|¬R1|)|4|≤m(m)m|gα_↓|g1|4|πln{H11}({H9}1/(1|4α_↓|4pz|gm){
 1114  H11}){H9}, |πfrom which we obtain |εa|βn|βp|4α=↓|4|¬K|ur|)d|
 1119  ¬Dn|)|4|≤m(n/d)n|gα_↓|g1p|gd; |πlim|ur|)|εp|¬M|¬X|)|4a|βn|βp
 1120  /p|gn|4α=↓|41/n. |πTo prove the stated idnetity, 
 1126  note that |¬K|ur|)|εn,j|¬R1|4|≤m(n)g(z|gn|gj)n|gα_↓|gtj|gα_↓
 1128  |gt|4α=↓|4|¬K|ur|)m|¬R1|)|4g(z|gm)m|gα_↓|gt|4|¬K|ur|)n|¬Dm|)
 1128  |4|≤m(n)|4α=↓|4g(z).|'{A3}|5|5|≡5|≡.|9|4|πLet 
 1130  |εa|βn|βp|βr |πbe the number of monic polynomials 
 1137  of degree |εn |πmodulo |εp |πhaving exactly |εr 
 1145  |πirreducible factors. Then |ε|λG|βp(z,|4w)|4α=↓|4|¬K|ur|)n,
 1148  r|¬R0|)|4a|βn|βp|βrz|gnw|gr|4α=↓|4|πexp{H11}({H9}|ε|¬K|ε|βk|
 1148  β|¬R|β1|4G|βp(z|gk)w|gk/k{H11}){H9}; |πcf. Eq. 
 1151  1.2.9<34. We have |¬K|ur|)|εn|¬R0|)|4A|βn|βpz|gn|4α=↓|4d|λG|
 1154  βp(z/p,|4w)/dw|4|¬,|βw|βα=↓|β1|4α=↓|4{H11}({H9}|¬K|ur|)k|¬R1
 1154  |)G|βp({H9}z|gk/p|gk){H11}){H9}|πexp{H11}({H9}ln(1/(1|4α_↓|4
 1154  |εz)){H11}){H9}|4α=↓|4(|¬K|ur|)|εn|¬R1|)|4|πln{H11}({H9}1/(1
 1154  |4α_↓|4|εp|g1|gα_↓|gnz|gn))|≤'(n)/n)/z{H11}){H9}, 
 1155  |πhence |εA|βn|βp|4α=↓|4H|βn|4α+↓|41/2p|4α+↓|4O(p|gα_↓|g2) 
 1157  |πfor |εn|4|¬R|42. |πThe average value of |ε2|gr 
 1164  |πis the coe∃cient of |εz|gn |πin |ε|λG|βp(z/p,|42), 
 1171  |πnamely |εn|4α+↓|41|4α+↓|4(n|4α+↓|41)/p|4α+↓|4O(p|gα_↓|g2).
 1172   (|πThe variance is of order |εn|g3, |πhowever: 
 1180  set |εw|4α=↓|44.)|'{A3}|5|5|≡6|≡.|9|4|πFor |ε0|4|¬E|4s|4|¬W|
 1183  4p, x|4α_↓|4s |πis a factor of |εx|gp|4α_↓|4x 
 1190  (|πmodulo |εp) |πby Fermat's theorem. So |εx|gp|4α_↓|4x 
 1197  |πis a multiple of lcm{H11}({H9}|εx|4α_↓|40, 
 1202  x|4α_↓|41,|4.|4.|4.|4,|4x|4α_↓|4(p|4α_↓|41){J11}){H9}|4α=↓|4
 1202  x|gp. (Note|*/: |\|πTherefore the Stirling numbers 
 1208  [|ε|fp|d5k|)] |πare multiples of |εp |πexcept 
 1214  when |εk|4α=↓|41, k|4α=↓|4p. |πEquation 1.2.6<41 
 1219  shows that the same statement is valid for Stirling 
 1228  numbers |¬A|ε|fp|d5k|)|¬S |πof the other kind.)|'
 1234  {A3}|5|5|≡7|≡.|9|4The factors on the right are 
 1240  relatively prime, and each is a divisor of |εu(x), 
 1249  |πso their product divides |εu(x). |πOn the other 
 1257  hand,|'{A6}|εu(x)!!|πdivides!!|εv(x)|gp|4α_↓|4v(x)|4α=↓|4|≥u
 1258  |uc|)0|¬Es|¬Wp|)|4{H11}({H9}v(x)|4α_↓|4s{H11}){H9},|;
 1259  {A6}|πso it divides the right-hand side by exercise 
 1267  4.5.2<2.|'{A3}|5|5|≡8|≡.|9|4The vector which 
 1271  is output in step N3 is the only output whose 
 1281  |εk|πth component is nonzero.|'{A3}|5|5|≡9|≡.|9|4For 
 1286  example, start with |εx|5|¬L|51, y|5|¬L|51; |πthen 
 1292  repeatedly set |εR[x]|5|¬L|5y, x|5|¬L2x|4|πmod|4101, 
 1296  |εy|5|¬L|551y|4|πmod|4101, one hundred times.|'
 1300  {A3}|≡1|≡0|≡.|9|4The matrix |εQ|4α_↓|4I |πbelow 
 1304  has a null space generated by the two vectors 
 1313  |εv|g[|g1|g]|4α=↓(1, 0, 0, 0, 0, 0, 0, 0), v|g[|g2|g]|4α=↓|4
 1321  (0, 1, 1, 0, 0, 1, 1, 1). |πThe factorization 
 1331  is|'{A6}(|εx|g6|4α+↓|4x|g5|4α+↓|4x|g4|4α+↓|4x|4α+↓|41)(x|g2|
 1332  4α+↓|4x|4α+↓|41).|;{A6}{H7L9}|∂!!!!|9|5|∂!!!!!!!!!!!!!!!!!!|
 1333  5|∂!!!!!!|∂!!!!!!!!!!!!!!!|9|6|∂!!!!|9|5|∂|E|;
 1334  |>|;|εp|4α=↓|42|;|;p|4α=↓|45|;>{A6}!!!!|9|40!!0!!0!!0!!0!!0!
 1340  !0!!0!!!!!!0!!0!!0!!0!!0!!0!!0|'!!!!|9|40!!1!!1!!0!!0!!0!!0!
 1341  !0!!!!!!0!!4!!0!!0!!0!!1!!0|'!!!!|9|40!!0!!1!!0!!1!!0!!0!!0!
 1342  !!!!!0!!2!!2!!0!!4!!3!!4|'!!!!|9|40!!0!!0!!1!!0!!0!!1!!0!!!!
 1343  !!0!!1!!4!!4!!4!!2!!1|'!!!!|9|41!!0!!0!!1!!0!!0!!1!!0!!!!!!2
 1344  !!2!!2!!3!!4!!3!!2|'!!!!|9|41!!0!!1!!1!!1!!0!!0!!0!!!!!!0!!0
 1345  !!4!!0!!1!!3!!2|'!!!!|9|40!!0!!1!!0!!1!!1!!0!!1!!!!!!3!!0!!2
 1346  !!1!!4!!2!!1|'!!!!|9|41!!1!!0!!1!!1!!1!!0!!1|'
 1348  {A12}{H9L11}|≡1|≡1|≡.|9|4|πRemoving the trivial 
 1351  factor |εx, |πthe matrix |εQ|4α_↓|4I |πabove 
 1357  has a null space generated by (1, 0, 0, 0, 0, 
 1368  0, 0) and (0, 3, 1, 4, 1, 2, 1). The factorization 
 1380  is|'{A6}|εx(x|g2|4α+↓|43x|4α+↓|44)(x|g5|4α+↓|42x|g4|4α+↓|4x|
 1381  g3|4α+↓|4rx|g2|4α+↓|4x|4α+↓|43).|;{A6}|≡1|≡2|≡.|9|πIf 
 1383  |εp|4α=↓|42, (x|4α+↓|41)|g4|4α=↓|4x|g4|4α+↓|41. 
 1385  |πIf |εp|4α=↓|48k|4α+↓|41, Q|4α_↓|4I |πis the 
 1390  zero matrix, so there are four factors. For other 
 1399  values of |εp |πwe have|'{A6}{H9L11}!!!!!!!!|9|7|εp|4α=↓|48k
 1404  |4α+↓|43!!!!|5|5p|4α=↓|48k|4α+↓|45!!!!|5|5p|4α=↓|48k|4α+↓|47
 1404  |'{A6}{M32}|h|εQ|4α_↓|4I|4α=↓|4|8|8|n0!!|8|80!!|8|80!!|8|80!
 1405  |9|50!!|8|80!!0!!|8|80!|9|50!!|8|80!!|8|80!!|8|80|'
 1406  {A6}|εQ|4α_↓|4I|4α=↓|4{B6}|8|80!!|→α_↓1!!|8|80!!|8|81!|9|50!
 1406  !|→α_↓2!!0!!|8|80!|9|50!!|→α_↓1!!|8|80!!|→α_↓1|'
 1407  |hQ|4α_↓|4I|4α=↓|4|8|8|n0!!|8|80!!|→α_↓2!!|8|80!|9|50!!|8|80
 1407  !!0!!|8|80!|9|50!!|8|80!!|→α_↓2!!|8|80|'Q|4α_↓|4I|4α=↓|4|8|8
 1408  0!!|8|81!!|8|80!!|→α_↓1!|9|50!!|8|80!!0!!|8|8|→α_↓2!|9|50!!|
 1408  →α_↓1!!|8|80!!|→α_↓1|'{A6}{M29}|πHere |εQ|4α_↓|4I 
 1411  |πhas rank 2, so there are 4|4α_↓|42|4α=↓|42 
 1418  factors. (But it is easy to prove that |εx|g4|4α+↓|41 
 1427  |πis irreducible over the integers, since it 
 1434  has no linear factors and the coe∃cient of |εx 
 1443  |πin any factor of degree two must be less than 
 1453  or equal to 2|4in absolute value by exercise 
 1461  20. For all |εk|4|¬R|42, |πH. P. F. Swinnerton-Dyer 
 1469  has exhibited polynomials of degree |ε2|gk |πwhich 
 1476  are irreducible over the integers, but which 
 1483  split completely into linear and quadratic factors 
 1490  modulo every prime. For degree 8, his example 
 1498  is |εx|g8|4α_↓|416x|g6|4α+↓|488x|g4|4α+↓192x|g2|4α+↓|4144, 
 1500  |πhaving roots |ε{U0}{H9L11M29}|πW58320#Computer 
folio 810 galley 3
 1503  Programming!(Knuth/Addison-Wesley)!Answers!f.|4810!g.|43d|'
 1504  {A6}|π|≡1|≡6|≡.|9|4(a) Substitute polynomials 
 1507  modulo |εp |πfor integers, in the proof for |εn|4α=↓|41. 
 1516  |π(b) The proof in exercise 3.2.1.2<16 carries 
 1523  over to any _nite _eld. (c) Since |εx|4α=↓|4|≤j|gk 
 1531  |πfor some |εk, x|gp|in|4α=↓|4x |πin the _eld. 
 1538  Furthermore, the elements |εy |πwhich satisfy 
 1544  the equation |εy|gp|im|4α=↓|4y |πin the _eld 
 1550  are closed under addition, and closed under multiplication; 
 1558  so if |εx|gp|im|4α=↓|4x, |πthen |ε|≤j |π(being 
 1564  a polynomial in |εx |πwith integer coe∃cients) 
 1571  satis_es |ε|≤j|gp|im|4α=↓|4|≤j.|'{A3}|π|≡1|≡7|≡.|9|4If 
 1574  |ε|≤j |πis a primitive root, each nonzero element 
 1582  is some power of |ε|≤j. |πHence the order must 
 1591  be a divisor of 13|g2|4α_↓|41|4α=↓|42|g3|4αo↓|43|4αo↓|47, 
 1596  and |ε|≤'(f) |πelements have order |εf.|'{A6}|∂!!|9|∂!!!|9|∂
 1602  !!!|∂!!!|9|∂!!!|∂!!!|∂!!!|9|∂!!!|9|∂|E|;|>f|;
 1605  |≤'(f)|;f|;|≤'(f)|;f|;|≤'(f)|;f|;|≤'(f)|;>{A2}|>
 1614  1|;1|;|9|13|;2|;|9|17|;|9|16|;|9|121|;12|;>|>
 1624  2|;1|;|9|16|;2|;14|;|9|16|;|9|142|;12|;>|>4|;
 1635  2|;12|;4|;28|;12|;|9|184|;24|;>|>8|;4|;24|;8|;
 1648  56|;24|;168|;48|;>{A6}|π|≡1|≡9|≡.|9|4If |εu(x)|4α=↓|4v(x)w(x
 1654  ) |πwith deg(|εv)|πdeg(|εw)|4|¬R|41, |πthen |εu|βnX|gn|4|"o|
 1658  4v(x)w(x) (|πmodulo |εp). |πBy unique factorization 
 1664  modulo |εp, |πall but the{A3}|π|≡2|≡0|≡.|9|4(a) 
 1669  |ε|¬K|4(|≤au|βj|4α_↓|4u|βj|βα_↓|β1)(|=3|≤a|=3u|βj|4α_↓|4|=3u
 1669  |βj|βα_↓|β1)|4α=↓|4|¬K|4(u|βj|4α_↓|4|=3|≤au|βj|βα_↓|β1)(|=3u
 1669  |βj|4α_↓|4|≤a|=3u↓|β1). (|πb) We may assume that 
 1675  |εu|β0|4|=|↔6α=↓|40. |πLet |εm(u)|4α=↓|4|≥7|β1|β|¬E|βj|β|¬E|
 1677  βn|4|πmin(|ε1,|4|¬G|≤a|βj|¬G)|4α=↓|4u|β0/M(u)u|βn. 
 1678  |πWhenever |¬G|ε|≤a|βj|¬G|4|¬W|41, |πchange the 
 1682  factor |εx|4α_↓|4|≤a|βj |πto |ε|=3|≤a|βjx|4α_↓|41 
 1686  |πin |εu(x); |πthis doesn't a=ect |ε|¬Gu|¬G. 
 1692  |πNow looking at the leading and trailing coe∃cients 
 1700  only, we have |ε|¬Gu|¬G|g2|4|¬R|4|¬Gu|βn|¬G|g2m(u)|g2|4α+↓|4
 1703  |¬Gu|βn|¬G|g2M(u)|g2; |πhence we obtain the slightly 
 1709  stronger result |εM(u)|g2|4|¬E|4{H11}({H9}|¬Gu|¬G|g2|4α+↓|4(
 1711  |¬Gu|¬G|g4|4α_↓|44|¬Gu|β0u|βn|¬G|g2)|g|f1|d32|){H11}){H9}/2|
 1711  ¬Gu|βn|¬G|g2. |π(c) |εu|βj|4α=↓|4u|βm|4|¬K|4|≤a|βi|m1|4.|4.|
 1713  4.|4|≤a|βi|mm|mα_↓|mj, |πan elementary symmetric 
 1717  function, hence |ε|¬Hu|βj|¬G|4|¬E|4|¬Gu|βm|¬G|4|¬K|4|≤b|βi|m
 1719  1|4.|4.|4.|4|≤b|βi|mm|mα_↓|mj |πwhere |ε|≤b|βi|4α=↓|4|πmax(1
 1721  ,|4|ε|¬G|≤a|βi|¬G). |πWe complete the proof by 
 1727  showing that when 1|4|¬E|4|εx|β1,|4.|4.|4.|4,|4x|βn 
 1731  |πand |εx|β1|4.|4.|4.|4x|βn|4α=↓|4M, |πthe elementary 
 1735  symmetric function |ε|≤s|βn|βk|4α=↓|4|¬K|4x|βi|m1|4.|4.|4.|4
 1737  x|βi|mk |πis |→|¬E(|ε|fnα_↓1|d5kα_↓1|))M|4α+↓|4(|fnα_↓1|d5k|
 1739  )), |πthe value assumed when |εx|β1|4α=↓|4αo↓|4αo↓|4αo↓|4α=↓
 1744  |4x|βn|βα_↓|β1|4α=↓|41 |πand |εx|βn|4α=↓|4M. 
 1747  (|πFor if |εx|β1|4|¬E|4αo↓|4αo↓|4αo↓|4|¬E|4x|βn|4|¬W|4M, 
 1750  |πthe transformation |εx|βn|4|¬L|4x|βn|βα_↓|β1x|βn, 
 1753  x|βn|βα_↓|β1|4|¬L|41 |πincreases |ε|≤s|βn|βk 
 1756  |πby |ε|≤s|β(|βn|βα_↓|β2|β)|β(|βk|βα_↓|β1|β)(x|βn|4α_↓|41)(x
 1757  |βn|βα_↓|β1|4α_↓|41)|4|¬Q|40.) |π(d) |ε|¬Gv|βj|¬G|4|¬E|4|¬Gv
 1759  |βm|¬G{H11}({H9}(|fmα_↓1|d5j|))M(v)|4α+↓|4(|fmα_↓1|d5jα_↓1|)
 1759  ){H11}){H9}|4|¬E|4|¬GU|βn|¬G{H11}({H9}(|fmα_↓1|d5j|))M(u)|4α
 1759  +↓|4(|fmα_↓1|d5jα_↓1|)){H11}){H9} |πsince |ε|¬Gv|βm|¬G|4|¬E|
 1761  4|¬Gu|βn|¬G |πand |εM(v)|4|¬E|4M(u). [|πThese 
 1765  results are slight extensions of inequalities 
 1771  due to M. Mignotte, |εMath. Comp. |π|≡2|≡8 (1974), 
 1779  1153<1157.]|'{A3}|≡2|≡1|≡.|9|4(a) |¬j|ur1|)0|)|4(|εu|βne(n|≤
 1781  u)|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|4u|β0)(|=3u|βne(|→α_↓n|≤u)|4α+↓|
 1781  4αo↓|4αo↓|4αo↓|4α+↓|4|=3u|β0)|4d|≤u|4α=↓|4|¬Gu|βn|¬G|g2|4α+↓
 1781  |4αo↓|4αo↓|4αo↓|4α+↓|4|¬Gu|β0|¬G|g2 |πsince |ε|¬J|ur1|)0|)|4
 1783  e(j|≤u)e(|→α_↓k|≤u)|4d|≤u|4α=↓|4|≤d|βj|βk; |πnow 
 1785  use induction on |εt. |π(b) Since |ε|¬Gv|βj|¬G|4|¬E|4(|fm|d5
 1791  j|))M(v)|¬Gv|βm|¬G |πwe conclude that |ε|¬Gv|¬G|g2|4|¬E|4(|f
 1795  2m|d5m|))M(v)|g2|¬Gv|βm|¬G|g2. |πHence |ε|¬Gv|¬G|g2|¬Gw|¬G|g
 1797  2|4|¬E|4(|f2m|d5m|))(|f2k|d5k|))M(v)|g2M(w)|g2|¬Gv|βmw|βk|¬G
 1797  |g2|4α=↓|4f(m,|4k)M(u)|g2|¬Gu|βn|¬G|g2|4|¬E|4f(m,|4k)|¬Gu|¬G
 1797  |g2. [|πSlightly better values for |εf(m,|4k) 
 1803  |πare possible based on the more detailed information 
 1811  in exercise 20.] (c) The case |εt|4α=↓|43 |πsu∃ces 
 1819  to show how to get from |εt|4α_↓|41 |πto |εt. 
 1828  |πWhen |εt|4α=↓|42 |πwe have shown that, for 
 1835  all |ε|≤u|β1, |¬J|ur1|)0|)|4|¬J|ur1|)0|)|4|¬J|ur1|)0|)|4|¬J|
 1837  ur1|)0|)|4|¬Gv{H11}({H9}e(|≤u|β1), e(|≤f|β2), 
 1839  e(|≤f|β3){H11}){H9}|¬G|g2|4|¬Gw{H11}({H9}e(|≤u|β1), 
 1840  e(|≤c|β2), e(|≤c|β3){H11}){H9}|¬G|g2|4d|≤f|β2|4d|≤f|β3|4d|≤c
 1841  |β2|4d|≤c|β3|4|¬E|4f(m|β2,|4k|β2)f(m|β3,|4k|β3)|4|¬J|ur1|)0|
 1841  )|4|¬J|ur1|)0|)|4|¬GV{H11}({H9}e(|≤u|β1), e(|≤u|β2), 
 1843  e(|≤u|β3){H11}){H9}|¬G|g2|4|¬Gw{H11}({H9}e(|≤u|β1), 
 1844  e(|≤u|β2), e(|≤u|β3){H11}){H9}|¬G|g2|4d|≤u|β2|4d|≤u|β3. 
 1846  |πFor all |ε|≤f|β2, |≤f|β3, |≤c|β2, |≤c|β3 |πwe 
 1853  have also shown that |ε|¬J|ur1|)0|)|4|¬J|ur1|)0|)|4|¬GV{H11}
 1857  ({H9}e(|≤f|β1), e(|≤f|β2), e(|≤f|β3){H11}){H9}|¬G|g2|4|¬Gw{H
 1859  11}({H9}e(|≤c|β1), e(|≤c|β2), e(|≤c|β3){H11}){H9}|¬G|g2|4d|≤
 1861  f|β1|4d|≤c|β1|4|¬E|4f(m|β1,|4k|β1)|4|¬J|ur1|)0|)|4|¬Gv{H11}(
 1861  {H9}e(|≤u|β1), e(|≤f|β2), e(|≤f|β3){H11}){H9}|¬G|g2|4|¬Gw{H1
 1863  1}({H9}e(|≤u|β1), e(|≤c|β2), e(|≤c|β3){H11}){H9}|¬G|g2|4d|≤u
 1865  |β1. |πIntegrate the former inequality wi7h respect 
 1872  to |ε|≤u|β1 |πand the latter with respect to 
 1880  |ε|≤f|β2, |≤f|β3, |≤c|β2, |≤c|β3. [|πThis method 
 1886  was used by A. O. Gel'fond in |εTranscendental 
 1894  and Algebraic Numbers (|πNew York: Dover, 1960), 
 1901  Section 3.4, to derive a slightly di=erent result.]|'
 1909  {A3}|≡2|≡2|≡.|9|4More generally, assume that 
 1913  |εu(x)|4|"o|4v(x)w(x) (|πmodulo |εq), a(x)v(x)|4α+↓|4b(x)w(x
 1916  )|4|"o|41 (|πmodulo |εp), |πand |εc|λ3(v)|4|"o|41 
 1921  (|πmodulo |εr), |πdeg(|εa)|4|¬W|4|πdeg(|εw), 
 1924  |πdeg(|εh)|4|¬W|4|πdeg(|εv), |πdeg(|εu)|4α=↓|4|πdeg(|εv)|4α+
 1925  ↓|4|πdeg(|εw), |πwhere |εr|4α=↓|4|πgcd(|εp,|4q) 
 1928  |πand |εp,|4q |πneedn't be prime. We shall construct 
 1936  polynomials |εV(x)|4|"o|4v(x) |πand |εW(x)|4|"o|4w(x) 
 1940  (|πmodulo |εq) |πsuch that |εu(x)|4|"o|4V(x)W(x) 
 1945  (|πmodulo |εqr), |λ3(V)|4α=↓|4|λ3(v), |πdeg(|εV)|4α=↓|4|πdeg
 1948  (|εW)|4α=↓|4|πdeg(|εw); |πfurthermore, if |εr 
 1952  |πis prime, the results will be unique modulo 
 1960  |εqr.|'!!|1|1|πThe problem asks us to _nd |ε|=3v(x) 
 1968  |πand |ε|=3w(x) |πwith |εV(x)|4α=↓|4v(x)|4α+↓|4q|=3v(x), 
 1972  {H11}({H9}N(x)|4α=↓|4w(x)|4α+↓|4q|=3w(x), |πdeg(|ε|=3v)|4|¬W
 1973  |4|πdeg(|εv), |πdeg(|ε|=3w)|4|¬E|4|πdeg(|εw); 
 1975  |πand the other condition {H11}({H9}|εv(x)|4α+↓|4q|=3v(x){H1
 1979  1})({H9}w(x)|4α+↓|4q|=3w(x){H11}){H9}|4|"o|4u(x) 
 1980  (|πmodulo |εqr) |πis equivalent to |ε|=3w(x)v(x)|4α+↓|4|=3v(
 1985  x)w(x)|4|"o|4f(x) (|πmodulo |εr), |πwhere |εu(x)|4|"o|4v(x)w
 1989  (x)|4α+↓|4qf(x) (|πmodulo |εqr). |πWe have |ε{H11}({H9}a(x)f
 1994  (x)|4α+↓|4t(x)w(x){H11}){H9}v(x)|4α+↓|4{H11}({H9}b(x)f(x)|4α
 1994  _↓|4t(x)v(x){H11}){H9}w(x)|4|"o|4f(x) (|πmodulo 
 1996  |εr) |πfor all |εt(x). |πSince |λ3(|εv) |πhas 
 2003  an inverse modulo |εr, |πwe can _nd a quotient 
 2012  |εt(x) |πby Algorithm 4.6.1D such that deg(|εbf|4α_↓|4tv)|4|
 2018  ¬W|4|πdeg(|εv); |πfor this |εt(x), |πdeg(|εaf|4α+↓|4tw)|4|¬E
 2022  |4|πdeg(|εw), |πsince deg(|εf)|4|¬E|4|πdeg(|εu)|4α=↓|4|πdeg(
 2024  |εv)|4α+↓|4|πdeg(|εw). |πThus the desired solution 
 2029  is |ε|=3v(x)|4α=↓|4b(x)f(x)|4α_↓|4t(x)v(x)|4α=↓|4b(x)f(x)|πm
 2030  od|4|εv(x), |=3w(x)|4α=↓|4a(x)f(x)|4α+↓|4t(x)w(x). 
 2032  |πIf {H11}({H9}|ε|=∩|=3v(x),|4|=∩|=3w(x){H11}) 
 2034  |πis another solution, we have {H11}({H9}|ε|=3w(x)|4α_↓|4|=∩
 2039  |=3w(x){H11}){H9}v(x)|4|"o|4{H11}({H9}|=∩|=3v(x)|4α_↓|4|=3v(
 2039  x){H11}){H9}w(x) (|πmodulo |εr). |πThus if |εr 
 2045  |πis prime, |εv(x) |πmust divide |ε|=∩|=3v(x)|4α_↓|4|=3v(x);
 2050   |πbut deg(|ε|=∩|=3v|4α_↓|4|=3v)|4|¬W|4|πdeg(|εv), 
 2053  |πso |ε|=∩|=3v(x)|4α=↓|4|=3v(x) |πand |ε|=∩|=3w(x)|4α=↓|4|=3
 2056  w(x).|'!!|1|1|πFor |εp|4α=↓|42, |πthe factorization 
 2061  proceeds as follows (writing only the coe∃cients, 
 2068  and using bars for negative digits): Exercise 
 2075  10 says that |εv|β1(x)|4α=↓|4(|=∩1|4|=∩1|4|=∩1), 
 2079  |εw|β1(x)|4α=↓|4(|=∩1|4|=∩1|4|=∩1|40|40|4|=∩1|4|=∩1) 
 2080  |πin one-bit two's complement notation. Euclid's 
 2086  extended algorithm yields |εa(x)|4α=↓|4(1|40|40|40|40|41), 
 2090  b(x)|4α=↓|4(1|40). |πThe factor |εv(x)|4α=↓|4x|g2|4α+↓|4c|β1
 2093  x|4α+↓|4c|β0 |πmust have |ε|¬Gc|β1|¬G|4|¬E|4|"l1|4α+↓|4|¬H|v
 2096  4113|)|"L|4α=↓|411, |¬Gc|β0|¬G|4|¬E|410, |πby 
 2099  exercise 20. Three applications of Hensel's lemma 
 2106  yield |εv|β4(x)|4α=↓|4(1|43|4|=∩1), w|β4(x)|4α=↓|4(1|4|=∩3|4
 2108  |=∩5|4|=∩4|44|4|=∩3|45). |πThus |εc|β1|4|"o|43 
 2111  |πand |εc|β0|4|"o|4|→α_↓1 (|πmodulo 16); the 
 2116  only possible quadratic factor of |εu(x) |πis 
 2123  |εx|g2|4α+↓|43x|4α_↓|41. |πDivision fails, so 
 2127  |εu(x) |πis irreducible. (Since we have now proved 
 2135  the irreducibility of this beloved polynomial 
 2141  by four separate methods, it is unlikely that 
 2149  it has any factors.)|'!!|1|1Hans Zassenhaus has 
 2156  observed that we can often speed up such calculations 
 2165  by increasing |εp |πas well as |εq: |πIn the 
 2174  above notation, we can _nd |εA(x), B(x) |πsuch 
 2182  that |εA(x)V(x)|4α+↓|4B(x)W(x)|4|"o|41 (|πmodulo 
 2185  |εpr), |πnamely by taking |εA(x)|4α=↓|4a(x)|4α+↓|4p|=3a(x), 
 2190  B(x)|4α=↓|4b(x)|4α+↓|4p|=∩b(x), |πwhere |ε|=3a(x)V(x)|4α+↓|4
 2192  |=∩b(x)W(x)|4|"o|4g(x) (|πmodulo |εr), a(x)V(x)|4α+↓|4b(x)W(
 2195  x)|4|"o|41|4α_↓|4pg(x) (|πmodulo |εpr). |πWe 
 2199  can also _nd |εC |πwith |λ3|ε(V)C|4|"o|41 (|πmodulo 
 2206  |εpr). |πIn this way we can lift a squarefree 
 2215  factorization |εu(x)|4|"o|4v(x)w(x) (|πmodulo 
 2218  |εp) |πto its unique extensions modulo |εp|g2, 
 2225  p|g4, p|g8, p|g1|g6, |πetc. However, this ``accelerated'' 
 2232  procedure reaches a point of diminishing returns 
 2239  in practice, as soon as we get to double-precision 
 2248  moduli, since the time for multiplying multiprecision 
 2255  numbers in practical ranges outweighs the advantage 
 2262  of squaring the modulus directly. From a computational 
 2270  standpoint it seems best to work with the successive 
 2279  moduli |εp| p|g2, p|g4, p|g8,|4.|4.|4.|4, p|gE, 
 2284  p|gE|gα+↓|ge, p|gE|gα+↓|g2|ge, p|gE|gα+↓|g3|ge,|4.|4.|4.|4, 
 2287  |πwhere |εE |πis the smallest power of 2 with 
 2296  |εp|gE |πgreater than single precision and |εe 
 2303  |πis the largest integer such that |εp|ge |πhas 
 2311  single precision.|'!!|1|1Hensel's Lemma, which 
 2316  he introduced in order to demonstrate the factorization 
 2324  of polynomials over the _eld of |εp-|πaddic numbers 
 2332  (see exercise 4.1<31), can be generalized in 
 2339  several ways. First, if there are more factors, 
 2347  say |εu(x)|4|"o|4v|β1(x)v|β2(x)v|β3(x) (|πmodulo 
 2350  |εp), |πwe can _nd |εa|β1(x), a|β2(x), a|β3(x) 
 2357  |πsuch that |εa|β1(x)v|β2(x)v|β3(x)|4α+↓|4a|β2(x)v|β1(x)v|β3
 2359  (x)|4α+↓|4a|β3(x)v|β1(x)v|β2(x)|4|"o|41 (|πmodulo 
 2361  |εp), |πdeg(|εa|βi)|4|¬W|4|πdeg(|εv|βi). (|πIn 
 2364  essence, |ε1/u(x) |πis expanded in partial fractions 
 2371  as |ε|¬K|4a|βi(x)/v|βi(x).) |πAn exactly analogous 
 2376  construction now allows us to lift the factorization 
 2384  without changing the leading coe∃cients of |εv|β1 
 2391  |πand |εv|β2; |πwe take |ε|=3v|β1(x)|4α=↓|4a|β1(x)f(x)|πmod|
 2395  4|εv|β1(x), |=3v|β2(x)|4α=↓|4a|β2(x)f(x)|πmod|4|εv|β2(x), 
 2397  |πetc. Another important generalization is to 
 2403  several simultaneous moduli, of the respective 
 2409  forms |εp|ge, (x|β2|4α_↓|4a|β2)|gn|r2,|4.|4.|4.|4, 
 2412  (x|βt|4α_↓|4a|βt)|gn|rt, |πwhen performing multivariate 
 2416  gcds and factorizations. Cf. D. Y. Y. Yun, Ph.D. 
 2425  Thesis (M.I.T., 1974).|'{A3}|≡2|≡3|≡.|9|4The 
 2429  discriminant of |εu(x) |πis a nonzero integer 
 2436  (cf. exercise 4.6.1<12), and there are multipoe 
 2443  factors modulo |εp |πi= |εp |πdivides the discriminant. 
 2451  (The factorization of (21) modulo 3 is (|εx|4α+↓|41)(x|g2|4α
 2458  _↓|4x|4α_↓|41)|g2(x|g3|4α+↓|4x|g2|4α_↓|4x|4α+↓|41); 
 2459  |πsquared factors for this polynomial occur only 
 2466  for |εp|4α=↓|43, 23, 233 |πand 121702457. It 
 2473  is not di∃cult to prove that the smallest prime 
 2482  which is not unlucky is at most |εO(n|4|πlog|4|εNn), 
 2490  |πif |εn|4α=↓|4|πdeg(|εu) |πand |εN |πbounds 
 2495  the coe∃cients of |εu(x).)|'{A12}|πCOMP: PLEASE 
 2501  PUT NUMBER 18 IN CORRECT POSITION|'{A6}|≡1|≡8.|9|4(a) 
 2508  pp{H11}({H9}|εp|β1(u|βnx){H11}){H9}|4.|4.|4.|4|πpp{H11}({H9}
 2508  |εp|βr(u|βnx){H11}){H9}, |πby Gauss's lemma. 
 2512  For example, let|'{A9}|εu(x)|4α=↓|46x|g3|4α_↓|43x|g2|4α+↓|42
 2515  x|4α_↓|41,!!v(x)|4α=↓|4x|g3|4α_↓|43x|g2|4α+↓|412x|4α_↓|436|4
 2515  α=↓|4(x|g2|4α+↓|412)(x|4α_↓|43);|;{A9}|πthen 
 2517  pp(|ε36x|g2|4α+↓|412)|4α=↓|43x|g2|4α+↓|41, |πpp(|ε6x|4α_↓|43
 2518  )|4α=↓|42x|4α_↓|41. (|πThis is a modern version 
 2524  of a fourteenth-century trick used for many years 
 2532  to help solve algebraic equations.)|'!!|1|1(b) 
 2538  Let pp{H11}({H9}|εw(u|βnx){H11}){H9}|4α=↓|4|=3w|βmx|gm|4α+↓|
 2539  4αo↓|4αo↓|4αo↓|4α+↓|4|=3w|β0|4α=↓|4w(u|βnx)/c, 
 2540  |πwhere |εc |πis the content of |εw(u|βnx) |πas 
 2548  a polynomial in |εx. |πThen |εw(x)|4α=↓|4(c|=3w|βm/u|urm|)n|
 2553  )|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|4c|=3w|β0, |πhence 
 2555  |εc|=3w|βm|4α=↓|4u|urm|)n|); |πsince |ε|=3w|βm 
 2558  |πis a divisor of |εu|βn, c |πis a multiple of 
 2568  |εu|urmα_↓1|)n|).|'|Hu*?{U31}TEAM|91|'{H9L11M29}W58320#Comput
folio 818 galley 4
 2570  er Programming|9(Knuth/Addison-Wesley)|9Answers|9f.|4818|9g.
 2571  |44d|'{A6}|≡2|≡4|≡.|9|4|πMultiply a monic polynomial 
 2576  with rational coe∃cients by a suitable nonzero 
 2583  integer, to get a primitive polynomial over the 
 2591  integers. Factor this polynomial over the integers, 
 2598  and then convert the factors back to monic. (No 
 2607  factorizations are lost in this way; see exercise 
 2615  4.6.1<8.)|'{A3}|≡2|≡5|≡.|9|4Consideration of 
 2618  the constant term shows there are no factors 
 2626  of degree 1, so if the polynomial is reducible, 
 2635  it must have one factor of degree 2 and one of 
 2646  degree 3. Modulo 2 the factors are |εx(x|4α+↓|41)|g2(x|g2α+↓
 2653  xα+↓1); |πthis is not much help. Modulo 3 the 
 2662  factors are (|εx|4α+↓|42)|g2(x|g3α+↓2xα+↓2). 
 2665  |πModulo 5 they are (|εx|g2|4α+↓|4x|4α+↓|41)(x|g3|4α+↓|44x|4
 2669  α+↓|42). |πSo we see that the answer is (|εx|g2|4α+↓|4x|4α+↓
 2677  |41)(x|g3|4α_↓|4x|4α+↓|42).|'{A3}|≡2|≡6|≡.|9|4|πBegin 
 2679  with |εD|2|3|¬L|5(0|4.|4.|4.|40|41), |πrepresenting 
 2682  the set |¬T0|¬Y. Then for 1|4|¬E|4|εj|4|¬E|4r, 
 2688  |πset |εD|5|¬L|5D|4|↔I|4(D |πshifted left |εd|βj), 
 2693  |πwhere |→|↔I denotes logical ``or.'' (Actually 
 2699  we need only |"p(|εn|4α+↓|41)/2|"P |πbits, since 
 2705  |εn|4α_↓|4j |πis in the set i= |εj |πis.)|'{A3}|≡2|≡7|≡.|9|4
 2713  |πExercise 4 says that a random polynomial of 
 2721  degree |εn |πis irreducible modulo |εp |πwith 
 2728  rather low probability, about |ε1/n. |πBut the 
 2735  Chinese remainder theorem implies that a random 
 2742  monic polynomial of degree |εn |πover the integers 
 2750  will be reducible with respect to each of |εk 
 2759  |πdistinct primes with probability about (|ε1|4α_↓|41/n)|gk,
 2764   |πand this approaches zero as |εk|5|¬M|5|→|¬X. 
 2771  |πHence almost all polynomials over the integers 
 2778  are irreducible with respect to in_nitely many 
 2785  primes; and almost all primitive polynomials 
 2791  over the integers are irreducible. [Another proof 
 2798  has been given by W. S. Brown, |εAMM |≡7|≡0 (1963), 
 2808  965<969.]|'{A3}|≡2|≡8|≡.|9|4False, we lose all 
 2813  |εp|βj |πwith |εe|βj |πdivisible by |εp. |πTrue 
 2820  if |εp|4|¬R|4|πdeg(|εu).|'{A3}|π|≡2|≡9|≡.|9|4Compute 
 2823  |εV|β1(x)|4α=↓|4|πgcd{H11}({H9}v(x),|4v|¬S(x){H11}){H9}; 
 2824  |πthis is legitimate since |εv|β1(x) |πis relatively 
 2831  prime to |εv|¬S(x)/v|β1(x). |πLet |εv|β0(x)|4α=↓|4v(x)/v|β1(
 2835  x), |πthe (squarefree) product of all irreducible 
 2842  factors of |εv(x). |πCompute |εd|β1(x)|4α=↓|4|πgcd{H11}({H9}
 2846  u(x),|4v|β0(x){H11}){H9} |πand |εu|β1(x)|4α=↓|4u(x)/d|β1(x).
 2848   |πIf deg(|εd|βj)|4|¬Q|40 |πfor |εj|4|¬R|41, 
 2853  |πcompute |ε|=∩d|βj|βα+↓|β1(x)|4α=↓|4|πgcd{H11}({H9}|εd|βj(x
 2854  ),|4v|βj(x){H11}){H9}, d|βj|βα+↓|β1(x)|4α=↓|4|πgcd{H11}({H9}
 2855  |ε|=∩d|βj|βα+↓|β1(x),|4u|βj(x){H11}){H9}, v|βj|βα+↓|β1(x)|4α
 2856  =↓|4v|βj(x)/|=∩d|βj|βα+↓|β1(x), u|βj|βα+↓|β1(x)|4α=↓|4u|βj(x
 2857  )/d|βj|βα+↓|β1(x); |πbut if deg(|εd|β1)|4α=↓|40, 
 2861  |π{H9}terminate the computation with the answer 
 2867  |εd(x)|4α=↓|4d|β1(x)|4.|4.|4.|4d|βj(x). {H11}({H9}|πIn 
 2869  this method, |εd|βj(x) |πis the squarefree product 
 2876  of all irreducible factors that occur |ε|→|¬Rj 
 2883  |πtimes in gcd({H9}|εu(x),|4v(x)){H9}. |πThere 
 2887  are several ways to avoid redundant calculations; 
 2894  for example, the same |εp |πcan be used in the 
 2904  underlying gcd computations, and the gcd routine 
 2911  should return the values of |εu(x)/d(x) |πand 
 2918  |εv(x)/d(x) |πthat it computes. Furthermore the 
 2924  computation is unsymmetric in |εu |πand |εv; 
 2931  |πit seems better to interchange the r|=7oles 
 2938  of |εu |πand |εv |πif deg(|εu)|4|¬Q|4|πdeg(|εv) 
 2944  |πat the beginning or if deg(|εu|βj)|4|¬Q|4|πdeg(|εv|βj) 
 2950  |πwhen computing |εd|βj|βα+↓|β1(x).{H11}){H9}|'
 2953  {A3}|≡3|≡0|≡.|9|4|πCf. exercise 4; the probability 
 2958  is the coe∃cient of |εz|gn |πin (|ε1α+↓a|β1|βpz/p)|→|6α⊗↓>
 2965  (1|3α+↓|3|εa|β2|βpz|g2/p|g2)(1|3α+↓|3a|β3|βpz|g3/p|g3)|4.|4.
 2965  |4.|4, |πwhich has the limiting value |εg(z)|3α=↓|3(1|3α+↓|3
 2971  |εz)(1|3α+↓|3|f1|d32|)z|g2)|→|5α⊗↓>(|ε1|4α+↓|4|f1|d33|)z|g3)
 2972  . |πFor |ε1|4|¬E|4n|4|¬E|410 |πthe answers are 
 2978  1, |f1|d32|), |f5|d36|), |f7|d312|), |f37|d360|), 
 2983  |f79|d3120|), |f173|d3280|), |f101|d3168|), |f127|d3210, 
 2987  |f1033|d31680|). {H11}({H9}|πLet>|εf(y)|4α=↓|4|πln(|ε1|4α+↓|
 2989  4y)|4α_↓|4y|4α=↓|4O(y|g1). |πNow |εg(z)|4α=↓|4|πexp{H11}({H9
 2991  }|¬K|ε|βn|β|¬R|β1|4z|gn/n|4α+↓|4|¬K|βn|β|¬R|β1|4f(z|gn/n){H1
 2991  1}){H9}|6|→α=↓>|εh(z)/(1|4α_↓|4z), |πnow it can 
 2996  be shown that the limiting probability is |εh(1)|6|→α=↓>
 3004  |πexp{H11}({H9}|ε|¬K|βn|β|¬R|β1|4f(1/n){H11}){H9}α=↓e|gα_↓|g
 3004  |≤g|4|¬V|4.56146 |πas |εn|5|¬M|5|→|¬X [|πcf. 
 3008  D. H. Lehmer, |εActa Arith. |π|≡2|≡1 (1972),>
 3015  |π379<388]. Indeed, N. G. de Bruijn has established 
 3023  the asymptotic formula lim|ε|βp|β|¬M|β|¬X|3a|βn|βp|6|→α=↓>
 3027  e|gα_↓|g|≤g|4α+↓|4e|gα_↓|g|≤g/n|4α+↓|4O(n|gα_↓|g2).|'
 3028  {A24}{H10L12}|∨S|∨E|∨C|∨T|∨I|∨O|∨N|9|∨4|∨.|∨6|∨.|∨3|'
 3029  {A12}{H9L11}|5|5|≡1|≡.|9|4|ε2|g|≤l|g(|gn|g), 
 3030  |πthe highest power of 2 less than or equal to 
 3040  |εn.|'{A3}|π|5|5|≡2|≡.|9|4Assume that |εx |πis 
 3045  input in register A, and |εn |πin location |¬n|¬n; 
 3054  the output is in register X.|'{A6}!!!|*/|↔c|↔O!!|\|¬a|¬1!|¬e|
 3060  ¬n|¬t|¬x!|¬1|5|6!!|h|εL|4α+↓|4|n1|h|4α_↓|4K!|nA|*/|↔O|\.|4|4I
 3060  nitialize.|'!!!|*/|↔c|↔P!!!|4!|\|¬s|¬t|¬x|5|6!|¬y|5|6!!|hL|4α
 3061  +↓|4|n1|h|4α_↓|4K!!|nY|5|¬L|51.|'!!!|*/|↔c|↔L!!|\!|4!|¬s|¬t|¬
 3062  a|5|6!|¬z!!|hL|4α+↓|4|n1|h|4α_↓|4K!!|nZ|5|¬L|5x.|'
 3063  !!!|*/|↔c|↔M|\!!!|4!|¬l|¬d|¬a|5|6!|¬n|¬n!!|hL|4α+↓|4|n1|h|4α_
 3063  ↓|4K!!|nN|5|¬L|5n.|'!!!|*/|↔c|↔C|\!!!|4!|¬j|¬m|¬p|5|6!|¬2|¬f!
 3064  !|hL|4α+↓|4|n1|h|4α_↓|4K!!|n|πTo|6A2.|'!!!|ε|*/|↔c|↔o|\!!|¬5|
 3065  ¬h!|¬s|¬r|¬b|5|6|¬1|5|6!!L|4α+↓|41|4α_↓|4K!!|'
 3066  !!!|*/|↔c|↔p!!|\!|4!|¬s|¬t|¬a|5|6!|¬n|5|6!!L|4α+↓|41|4α_↓|4K!
 3066  !N|5|¬M|5|"lN/2|"L.|'!!!|*/|↔c|↔l|\!!|π|¬a|¬5!|¬l|¬d|¬a|5|6!|
 3067  ¬z|5|6!!|hL|4α+↓|3|n|εL|h|3α_↓|4K!!|nA|*/|↔C|6|\Square|6Z.|'
 3068  !!!|*/|↔c|↔m|\!!!|4!|π|¬m|¬u|¬l|5|6!|¬z|5|6!!|h|εL|4α+↓|3|nL|
 3068  h|3α_↓|4K!!|nZ|4α⊗↓|4Z|'!!!|ε|*/|↔O|↔c|\!!!|4!|π|¬s|¬t|¬x|5|6
 3069  !|¬z|5|6!!|h|εL|4α+↓|3|nL|h|3α_↓|4K!!|n!|¬M|5Z.|'
 3070  !!!|*/|↔O|↔O|\!!|π|¬a|¬2!|¬l|¬d|¬a|5|6|¬n|5|6!!|h|εL|4α+↓|3|n
 3070  L|h|3α_↓|4K!!|nA|*/|↔P|\.|4|4Halve|6N.|'!!!|*/|↔O|↔P!!|\|π|¬2|
 3071  ¬h!|¬j|¬a|¬e|5|6!|¬5|¬b!!|h|ε|2α_↓|nL|4α+↓|41|h|6K!!|n|πTo|6
 3071  A5|6if|6|εN|6|πis|6even.|'!!!|*/|↔O|↔L!!!|4!|\|π|¬s|¬r|¬b|5|6
 3072  !|¬1|5|6!!|h|εL|7α+↓|nK|h|91α+↓|n|'!!!|*/|↔O|↔M!!|\!|4!|π|¬s|
 3073  ¬t|¬a|5|6!|¬n|5|6!!|h|ε|7Lα+↓|nK|h|91α+↓!!|nN|5|¬M|5|"lN/2|"
 3073  L.|'!!!|*/|↔O|↔C|\!!|π|¬a|¬3!|¬l|¬d|¬a|5|6!|¬z|5|6!!|h|εL|7α+
 3074  ↓|nK|h1|9α+↓!!|nA|*/|↔L|\.|4|4Multiply|6Y|6by|6Z.|'
 3075  !!!|*/|↔O|↔o!!!|4|\!|π|¬m|¬u|¬l|5|6!|¬y|5|6!!|h|εL|7α+↓|nK|h1
 3075  |9α+↓!!|nZ|4α⊗↓|4Y|'!!!|*/|↔O|↔p|\!!!|4!|π|¬s|¬t|¬x|5|6!|¬y|5
 3076  |6!!|h|εL|7α+↓|nK|h1|9α+↓!!!!|n|¬M|5Y.|'!!!|*/|↔O|↔l|\!!|π|¬a
 3077  |¬4!|¬l|¬d|¬a|5|6!|¬n|5|6!!|h|εL|7α+↓|nK|h1|9α+↓!!|nA|*/|↔M|\
 3077  .|4|4N|4α=↓|4|*/|↔c?|'!!!|↔O|↔m!!|\!|4!|π|¬j|¬a|¬p|5|6!|¬a|¬5
 3078  !!|h|εL|7α+↓|nK|h1|9α+↓!!|n|πIf|6not,|6continue|6at|6step|6A
 3078  5.|'{A6}{H9L11}{H11}({H9}|πIt would be better 
 3083  programming practice to change the instruction 
 3089  in line 05 to ``|¬j|¬a|¬p'', followed by an error 
 3098  indication. The running time is |ε21L|4α+↓|417K|4α+↓|413, 
 3104  |πwhere |εL|4α=↓|4|≤l(n) |πis one less than the 
 3111  number of bits in the binary representation of 
 3119  |εn, |πand |εK|4α=↓|4|≤n(n) |πis the number of 
 3126  one bits in |εn'|πs representation. The running 
 3133  time could be decreased by |εK|4α+↓|46 |πunits, 
 3140  by inserting step A4 before step A3 and adding 
 3149  two new instructions to perform A3 when |εN|4α=↓|40.{H11}){H
 3156  9}|'!!|2|πFor the serial program, we may assume 
 3164  that |εn |πis small enough to _t in an index 
 3174  register; otherwise serial exponentiation is 
 3179  out of the question. The following program leaves 
 3187  the output in register A:|'{A6}!!!!!|9|π|¬s|¬1!|¬l|¬d|¬1|5|6
 3192  !|¬n|5|6!!|h|εN|6|n1!|4|πrI1|5|¬M|5|εn.|'!!!!!|9!|4!|π|¬s|¬t
 3193  |¬a|5|6!|¬x|5|6!!|h|εN|6|n1!|4|εX|5|¬M|5x.|'|π!!!!!|9!|4!|¬j
 3194  |¬m|¬p|5|6!|¬2|¬f!!|h|εN|6|n1!|4|'!!!!!|9|π|¬1|¬h!|¬m|¬u|¬l|
 3195  5|6!|¬x|5|6!!|εN|4α_↓|41!!|πrA|4α⊗↓|4X|'!!!!!|9!|4!|π|¬s|¬l|
 3196  ¬a|¬x!|¬5!!|εN|4α_↓|41!!!!|6|¬M|5|πrA.|'!!!!!|9|¬2|¬h!|¬d|¬e
 3197  |¬c|¬1!|¬1|5|6!!|n|9|5|5|εN|9|5|5!!|πrI1|5|¬M|5rI1|4α_↓|41.|
 3197  '!!!!!|9!|4!|π|¬j|¬1|¬p|5|6!|¬1|¬b!!|9|5|5|εN|9|5|5!!|πMulti
 3198  ply|6again|6if|6rI1|4|¬Q|40.|'{A6}{H9L11M29}|πThe 
 3200  running time for this program is |ε14N|4α_↓|47; 
 3207  |πit is faster than the previous program when 
 3215  |εn|4|¬R|47, |πslower when |εn|4|¬R|48.|'|π|5|5|≡3|≡.|9|4|πT
 3219  he sequences of e ponents are: (a) 1, 2, 3, 6, 
 3230  7, 14, 15, 30, 60, 120, 121, 242, 243, 486, 487, 
 3241  974, 974 [16 multiplications]; (b) 1, 2, 3, 4, 
 3250  8, 12, 24, 36, 72, 108, 216, 324, 325, 650, 975 
 3261  [14 multiplications]; (c) 1, 2, 3, 6, 12, 15, 
 3270  30, 60, 120, 240, 243, 486, 972, 975 [13 multiplications]; 
 3280  (d) 1, 2, 3, 6, 13, 15, 30, 60, 75, 150, 225, 
 3292  450, 900, 975 [13 multiplications]. {H11}({H9}The 
 3298  fewest possible number of multiplications is 
 3304  12; this is obtainable by combining the factor 
 3312  method with the binary method, since 975|4α=↓15|4|¬O|4(2|g6|
 3318  4α+↓|41).{H11}){H9}|'|5|5|≡4|≡.|9|4(777777)|β8|4α=↓|42|g1|g8
 3319  |4α_↓|41.|'{A3}{I3.3H}|5|5|≡5|≡.|9|4|≡T|≡1|≡.|9|4[Initialize
 3320  .] Set |¬l|¬i|¬n|¬k|¬u[|εj]|π|5|¬M|50 for |ε1|4|¬E|4j|4|¬E|4
 3324  2|gr, |πand set |εk|5|¬M|50, |π|¬l|¬i|¬n|¬k|¬r[0]|5|¬M|51, 
 3329  |¬l|¬i|¬n|¬k|¬r[1]|5|¬M|50.|'{A3}!!|2|≡T|≡2|≡.|9|4Change 
 3331  level.] (Now level |εk |πof the tree has been 
 3340  linked together from left to right, starting 
 3347  at |¬l|¬i|¬n|¬k|¬r[0].) If |εk|4α=↓|4r, |πthe 
 3352  algorithm terminates. Otherwise set |εn|5|¬M|5|π|¬l|¬i|¬n|¬k
 3356  |¬r[0], |εm|5|¬M|50.|'{A3}!!|2|π|≡T|≡3|≡.|9|4[Prepare 
 3359  for |εn.] (|πNow |εn |πis a node on level |εk, 
 3369  |πand |εm |πpoints to the rightmost node currently 
 3377  on level |εk|4α+↓|41.) |πSet |εq|5|¬L|50, s|5|¬L|5n.|'
 3383  {A3}!!|2|π|≡T|≡4|≡.|9|4[|πAlready in tree?] (Now 
 3387  |εs |πis a node in the path from the root to 
 3398  |εn.) |πIf |¬l|¬i|¬n|¬k|¬u[|εn|4α+↓|4s]|π|4|=|↔6α=↓|40, 
 3401  go to T6 (the value |εn|4α+↓|4s |πis already 
 3409  in the tree).|'{A3}!!|2|≡T|≡5|≡.|9|4[Insert below 
 3414  |εn.] |πIf |εq|4α=↓|40, |πset |εm|¬S|5|¬L|5n|4α+↓|4s. 
 3419  |πSet |¬l|¬i|¬n|¬k|¬r[|εn|4α+↓|4s]|5|¬L|5q, |π|¬l|¬i|¬n|¬k|¬
 3421  u[|εn|4α+↓|4s]|5|¬L|5n, q|5|¬L|5n|4α+↓|4s.|'{A3}!!|2|π|≡T|≡6
 3423  |≡.|9|4[Move up.] Set |εs|5|¬L|5|π|¬l|¬i|¬n|¬k|¬u[|εs]. 
 3427  |πIf |εs|4|=|↔6α=↓|40, |πreturn to T4.|'{A3}!!|2|≡T|≡7|≡.|9|
 3432  4[Attach group.] If |εq|4|=|↔6α=↓|40, |πset |¬l|¬i|¬n|¬k|¬r[
 3437  |εm]|5|¬L|5q, m|5|¬L|5m|¬S.|'{A3}!!|2|π|≡T|≡8|≡.|9|4[Move 
 3440  |εn.] |πSet |εn|5|¬L|5|π|¬l|¬i|¬n|¬k|¬r[|εn]. 
 3443  |πIf |εn|4|=|↔6α=↓|40, |πreturn to T3.|'{A3}!!|2|≡T|≡9|≡.|9|
 3448  4[End of level.] Set |¬l|¬i|¬n|¬k|¬r[|εm]|5|¬L|50, 
 3453  k|5|¬L|5k|5α+↓|51, |πand return to T2.|'{A3}{IC}|5|5|≡6|≡.|9
 3458  |4Prove by induction that the path to the number 
 3467  |ε2|ge|r0|4α+↓|42|ge|r1|4α+↓|4|¬O|4|¬O|4|¬O|4α+↓|42|ge|rt, 
 3468  |πif |εe|β0|4|¬Q|4e|β1|4|¬Q|4|¬O|4|¬O|4|¬O|4|¬Qe|βt|4|¬R|40,
 3469   |πis |ε1, 2, 2|g2,|4.|4.|4.|4,|42|ge|r0, 2|ge|r0|4α+↓|42|ge
 3474  |r1, .|4.|4.|4, 2|ge|m0|4α+↓|42|ge|r1|4α+↓|4|¬O|4|¬O|4|¬O|4α
 3476  +↓|42|ge|rt; |πfurthermore, the sequence of exponents 
 3482  on each level are in decreasing lexicographic 
 3489  order.|'{A3}|5|5|≡7|≡.|9|4The binary and factor 
 3494  methods require one more step to compute |εx|g2|gn 
 3502  |πthan |εx|gn; |πthe power tree method requires 
 3509  at most one more step. hence (a) 15|4|¬O|42|ε|gk; 
 3517  (|πb) 33|4|¬O|4|ε2|gk; (|πc) 23|4|¬O|4|ε2|gk; 
 3521  k|4α=↓|40, 1, 2, 3,|4.|4.|4.|4.|'{A3}|5|5|≡8|≡.|9|4|πThe 
 3526  power tree always includes the node |ε2m |πat 
 3534  one level below |εm, |πunless it occurs at the 
 3543  same level or an earlier level; and it always 
 3552  includes the node |ε2m|4α+↓|41 |πat one level 
 3559  below |ε2m, |πunless it occurs at the same level. 
 3568  (Computational experiments have shown that |ε2m 
 3574  |πis below |εm |πfor all |εm|4|¬E|4|π2000, but 
 3581  it appears very di∃cult to prove this in general.)|'
 3590  {A3}|≡1|≡0|≡.|9|4By using the ``|¬f|¬a|¬t|¬h|¬e|¬r'' 
 3594  representation discussed in Section 2.3.3: Make 
 3600  use of a table |εf[j], 1|4|¬E|4J|4|¬E|4100, |πsuch 
 3607  that |εf[1]|4α=↓|40 |πand |εf[j] |πis the number 
 3614  of the node just above |εj |πfor |εj|4|¬R|42. 
 3622  (|πThe fact that each node of this tree has degree 
 3632  at most two has no e=ect on the e∃ciency of this 
 3643  representation; it just makes the tree look prettier 
 3651  as an illustration.)|'{A3}|≡1|≡1|≡.|9|41, 2, 
 3656  3, 5, 10, 20, (23 or 40), 43; 1, 2, 4, 8, 9, 
 3669  17, (26 or 34), 43; 1, 2, 4, 8, 9, 17, 34, (43 
 3682  or 68), 77; 1, 2, 4, 5, 9, 18, 36, (41 or 72), 
 3695  77. If either of the latter two paths were in 
 3705  the tree we would have no possibility for |εn|4α=↓|443, 
 3714  |πsince the tree must contain either 1, 2, 3, 
 3723  5 or 1, 2, 4, 8, 9.|'{A3}|≡1|≡2|≡.|9|4No such 
 3732  in_nite tree can exist since |εl(n)|4|=|↔6α=↓|4l|≤⊂(n) 
 3738  |πfor some |εn.|'|Hβ*?*?*?*?{U0}{H9L11M29}|πW58320#Computer|4Pro
folio 821 galley 5
 3741  gramming!(Knuth/Addison-Wesley)!Answers!f.|4821!g.|45d|'
 3742  {A6}|≡1|≡3|≡.|9|4For Case 1, use a Type-1 chain 
 3749  followed by |ε2|gA|gα+↓|gC|4α+↓|42|gB|gα+↓|gC|4α+↓|42|gA|4α+
 3751  ↓|42|gB; |πor use the factor method. For Case 
 3759  2, use a Type-2 chain followed by 2|ε|gA|gα+↓|gC|gα+↓|g1|4α+
 3766  ↓|42|gB|gα+↓|gC|4α+↓|42|gA|4α+↓|42|gB. |πFor 
 3768  Case 3, use a Type-5 chain followed by addition 
 3777  of |ε2|gA|4α+↓|42|gA|gα_↓|g1, |πor use the factor 
 3783  method. For Case 4, |εn|4α=↓|4135|4αo↓|42|gD, 
 3788  |πso we may use the factor method.|'{A3}|≡1|≡4|≡.|9|4a)|9It 
 3796  is easy to verify that steps |εr|4α_↓|41 |πand 
 3804  |εr|4α_↓|42 |πare not both small, so let us assume 
 3813  that step |εr|4α_↓|41 |πis small and step |εr|4α_↓|42 
 3821  |πis not. If |εc|4α=↓|41, |πthen |ε|≤l(a|βr|βα_↓|β1)|4α=↓|4|
 3826  ≤l(a|βr|βα_↓|βk), |πso |εk|4α=↓|42; |πand since 
 3831  |ε4|4|¬E|4|≤n(a|βr)|4α=↓|4|≤n(a|βr|βα_↓|β1)|4α+↓|4|≤n(a|βr|β
 3831  α_↓|βk)|4α_↓|41|4|¬E|4|≤n(a|βr|βα_↓|β1)|4α+↓|41, 
 3832  |πwe have |ε|≤n(a|βr|βα_↓|β1)|4|¬R|43, |πmaking 
 3836  |εr|4α_↓|41 |πa star step (lest |εa|β0, a|β1,|4.|4.|4.|4, 
 3843  a|βr|βα_↓|β3, a|βr|βα_↓|β1 |πinclude only one 
 3848  small step). Then |εa|βr|βα_↓|β1|4α=↓|4a|βr|βα_↓|β2|4α+↓|4a|
 3851  βr|βα_↓|βq |πfor some |εq, |πand if we replace 
 3859  |εa|βr|βα_↓|β2, a|βr|βα_↓|β1, a|βr |πby |εa|βr|βα_↓|β2, 
 3864  2a|βr|βα_↓|β2, 2a|βr|βα_↓|β2|4α+↓|4a|βr|βα_↓|βq|4α=↓|4a|βr, 
 3866  |πwe obtain another counterexample chain in which 
 3873  step |εr |πis small; but this is impossible. 
 3881  On the other hand, if |εc|4|¬R|42, |πthen 4|4|¬E|4|ε|≤n(a|βr
 3888  )|4|¬E|4|≤n(a|βr|βα_↓|β1)|4α+↓|4|≤n(a|βr|βα_↓|βk)|4α_↓|42|4|
 3888  ¬E|4|≤n(a|βr|βα_↓|β1); |πhence |ε|≤n(a|βr|βα_↓|β1)|4α=↓|44, 
 3891  |≤n(a|βr|βα_↓|βk)|4α=↓|42, |πand |εc|4α=↓|42. 
 3894  |πThis leads readily to an impossible situation 
 3901  by a consideration of the six types in the proof 
 3911  of Theorem B.|'!!|1|1b)|9If |ε|≤l(a|βr|βα_↓|βk)|4|¬W|4m|4α_↓
 3915  |41, |πwe have |εC|4|¬R|43, |πso |ε|≤n(a|βr|βα_↓|βk)|4α+↓|4|
 3920  ≤n(a|βr|βα_↓|β1)|4|¬R|47 |πby (22); therefore 
 3924  both |ε|≤n(a|βr|βα_↓|βk) |πand |ε|≤n(a|βr|βα_↓|β1) 
 3928  |πare |→|¬R3. |πAll small steps must be |ε|→|¬Er|4α_↓|4k, 
 3936  |πand |ε|≤l(a|βr|βα_↓|βk)|4α=↓|4m|4α_↓|4k|4α+↓|41. 
 3938  |πIf |εk|4|¬R|44, |πwe must have |εc|4α=↓|44, 
 3944  k|4α=↓|44, |≤n(a|βr|βα_↓|β1)|4α=↓|4|≤n(a|βr|βα_↓|β4)|4α=↓|44
 3945  ; |πthus |εa|βr|βα_↓|β1|4|¬R|42|gm|4α+↓|42|gm|gα_↓|g1|4α+↓|4
 3947  2|gm|gα_↓|g2, |πand |εa|βr|βα_↓|β1 |πmust equal 
 3952  |ε2|gm|4α+↓|41|gm|gα_↓|g1|4α+↓|42|gm|gα_↓|g2|4α+↓|42|gm|gα_↓
 3952  |g3; |πbut |εa|βr|βα_↓|β4|4|¬R|4|f1|d38|)a|βr|βα_↓|β1 
 3955  |πnow implies that |εa|βr|βα_↓|β1|4α=↓|48a|βr|βα_↓|β4. 
 3959  |πThus |εk|4α=↓|43 |πand |εa|βr|βα_↓|β1|4|¬Q|42|gm|4α+↓|42|g
 3962  m|gα_↓|g1. |πSince |εa|βr|βα_↓|β2|4|¬W|42|gm 
 3965  |πand |εa|gr|gα_↓|g3|4|¬W|42|gm|gα_↓|g1, |πstep 
 3968  |εr|4α_↓|41 |πmust be a doubling; but step |εr|4α_↓|42 
 3976  |πis a nondoubling, since |εa|βr|βα_↓|β1|4|=|↔6α=↓|44a|βr|βα
 3980  _↓|β3. |πFurthermore, since |ε|≤n(a|βr|βα_↓|β3)|4|¬R|43, 
 3984  r|4α_↓|43 |πis a star step; and |εa|βr|βα_↓|β2|4α=↓|4a|βr|βα
 3990  _↓|β3|4α+↓|4a|βr|βα_↓|β5 |πwould imply that |εa|βr|βα_↓|β5|4
 3994  α=↓|42|gm|gα_↓|g2, |πhence we must have |εa|βr|βα_↓|β2|4α=↓|
 3999  4a|βr|βα_↓|β3|4α+↓|4a|βr|βα_↓|β4. |πAs in a similar 
 4004  case treated in the text, the only possibility 
 4012  is now seen to be |εa|βr|βα_↓|β4|4α=↓|42|gm|gα_↓|g2|4α+↓|42|
 4017  gm|gα_↓|g3, a|βr|βα_↓|β3|4α=↓|42|gm|gα_↓|g2|4α+↓|42|gm|gα_↓|
 4018  g3|4α+↓|42|gd|gα+↓|g1|4α+↓|42|gd, a|βr|βα_↓|β1|4α=↓|42|gm|4α
 4019  +↓|42|gm|gα_↓|g1|4α+↓|42|gd|gα+↓|g2|4α+↓|42|gd|gα+↓|g1, 
 4020  |πand even this possibility is impossible.|'{A3}|≡1|≡6|≡.|9|
 4026  4|λ3|ε|gB(n)|4α=↓|4|≤l(n)|4α+↓|4|≤n(n)|4α_↓|41; 
 4027  |πso if |εn|4α=↓|42|gk, |λ3|gB(n)/|≤l(n)|4α=↓|41, 
 4031  |πbut if |εn|4α=↓|42|gk|gα+↓|g1|4α_↓|41, |λ3|gB(n)/|≤l(n)|4α
 4034  =↓|42.|'{A3}|≡1|≡7|≡.|9|4|πLet |εi|β1|4|¬W|4αo↓|4αo↓|4αo↓|4|
 4036  ¬W|4i|βt. |πDelete any intervals |εI|βk |πwhich 
 4042  can be removed without a=ecting the union |εI|β1|4|↔q|4αo↓|4
 4049  αo↓|4αo↓|4|↔q|4I|βt. (|πThe interval (|εj|βk,|4i|βk] 
 4053  |πmay be dropped out if either |εj|βk|βα+↓|β1|4|¬E|4j|βk 
 4060  |πor |εj|β1|4|¬W|4j|βk |πor |εj|β1|4|¬W|4j|β2|4|¬W|4αo↓|4αo↓
 4063  |4αo↓ |πand |εj|βk|βα+↓|β1|4|¬E|4i|βk|βα_↓|β1.) 
 4066  |πNow combine overlapping intervals (|εj|β1,|4i|β1],|4.|4.|4
 4070  .|4,|4(j|βd,|4i|βd] |πinto an interval (|εj|1|¬S,|4i|¬S]|4α=
 4074  ↓|4(j|β1,|4i|βd] |πand note that|'{A6}{A3}|εa|βi|β|¬S(1|4α+↓
 4078  |4|≤d)|gi|r1|iα_↓|ij|r1|gα+↓|1|gαo↓|1|gαo↓|1|gαo↓|1|gα+↓|gi|
 4078  rd|gα_↓|gj|rd|4|¬E|4a|βj|β|¬S(1|4α+↓|4|≤d)|g2|g(|gi|g|¬S|gα_
 4078  ↓|gj|g|¬S),|;{A9}|πsince each point of (|εj|1|¬S,|4i|¬S] 
 4084  |πis covered at most twice in (|εj|βi,|4i|β1]|4|↔q|4αo↓|4αo↓
 4090  |4αo↓|4|↔q|4(j|βd,|4i|βd].|'{A3}|π|≡1|≡8|≡.|9|4Call 
 4092  |εf(m) |πa ``nice'' function|=|ε|gm|¬H|v4f(m)|)|4|¬M|41 
 4096  |πas |εm|4|¬M|4|¬X, |πthat is, {H11}({H9}log|4|εf(m){H11}){H
 4100  9}/m|4|¬M|40. |πA polynomial in |εm |πis nice. 
 4107  The product of nice functions is nice. If |εg(m)|4|¬M|40 
 4116  |πand |εc |πis a positive constant, then |εc|gm|gg|g(|gm|g) 
 4124  |πis nice; also (|ε|f2m|d5mg(m)|) |πis nice, 
 4130  for by Stirling's approximation this is equivalent 
 4137  to saying that |εg(m)|4|πlog{H11}({H9}|ε1/g(m){H11}){H9}|4|¬
 4140  M|40.|'{A3}!!|1|1|πNow replace each term of the 
 4147  summation by the maximum term which is attained 
 4155  for any |εs, t, v. |πThe total number of terms 
 4165  is nice, and so are (|ε|fmα+↓s|d5tα+↓v|)), (|ftα+↓v|d5v|))|4
 4171  |¬E|42|gt|gα+↓|gv, |πand |ε|≤b|g2|rv, |πbecause 
 4175  (|εt|4α+↓|4v)/m|4|¬M|40. |πFinally, (|ε|f(mα+↓s)|i2|d5t|))|4
 4177  |¬E|4(2m)|g2|gt/t*3|4|¬W|4(4m|g2/t)|gte|gt, |πwhere 
 4179  (|ε4e)|gt |πis nice; setting |εt |πto its maximum 
 4187  value (1|4α_↓|4|ε|f1|d32|)|≤e)m/|≤l(m), |πwe 
 4190  have |ε(m|g2/t)|gt|4α=↓|4({H11}(m|≤l(m)/(1|4α_↓|4|f1|d32|)|≤
 4191  e){H11}){H9}|gt|4α=↓|42|gm|g(|g1|gα_↓|g|≤e|g/|g2|g)|4αo↓|4f(
 4191  m), |πwhere |εf(m) |πis nice. Hence the entire 
 4199  sum is less than |ε|≤a|gm |πfor large |εm, |πif 
 4208  |ε|≤a|4α=↓|42|g1|gα_↓|g|≤h, 0|4|¬W|4|≤h|4|¬W|4|f1|d32|)|≤e.|
 4209  '{A3}|≡1|≡9|≡.|9|4|πa)|9|1|εM|4|↔Q|4N, M|4|↔q|4N, 
 4212  M|4|=|rα+↓|↔q|4N, |πrespectively; see Eqs. 4.5.2<6, 
 4217  4.5.2<7.|'!!|1|1b)|9|εf(z)g(z), |πlcm{H11}({H9}|εf(z),|4g(z)
 4219  {H11}){H9}, |πgcd{H11}({H9}|εf(z),|4g(z){H11}){H9}. 
 4221  (|πFor the same reasons as (a), because the monic 
 4230  irreducible polynomials over the complex numbers 
 4236  are precisely the polynomials |εz|4α_↓|4|≤z.)|'
 4241  !!|1|1|πc)|9Commutative laws |εA|4|"b|4B|4α=↓|4B|4|"b|4A, 
 4244  A|4|↔q|4B|4α=↓|4B|4|↔q|4A, A|4|↔Q|4B|4α=↓|4B|4|↔Q|4A. 
 4246  |πAssociative laws |εA|4|"b|4(B|4|"b|4C)|4α=↓|4(A|4|"b|4B)|4
 4248  |"b|4C, A|4|↔q|4(B|4|↔q|4C)|4α=↓|4(A|4|↔q|4B)|4|↔q|4C, 
 4250  A|4|↔Q|4(B|4|↔Q|4C)|4α=↓|4(A|4|↔Q|4B)|4|↔Q|4C. 
 4251  |πDistributive laws |εA|4|↔q|4(B|4|↔Q|4C)|4α=↓|4(A|4|↔q|4B)|
 4253  4|↔Q|4(A|4|↔q|4C), A|4|↔Q|4(B|4|↔q|4C)|4α=↓|4(A|4|↔Q|4B)|4|↔
 4254  q|4(A|4|↔Q|4C), A|4|"b|4(B|4|↔q|4C)|4α=↓|4(A|4|"b|4B)|4|↔q|4
 4255  (A|4|"b|4C), A|4|"b|4(B|4|↔Q|4C)|4α=↓|4(A|4|"b|4B)|4|↔Q|4(A|
 4256  4|"b|4C). |πIdempotent laws |εA|4|↔q|4A|4α=↓|4A, 
 4260  a|4α=↓|4A. |πAbsorption laws |εA|4|↔q|4(A|4|↔Q|4B)|4α=↓|4A, 
 4264  A|4|↔Q|4(A|4|↔q|4B)|4α=↓|4A, A|4|↔Q|4(A|4|"b|4B)|4α=↓|4A, 
 4266  A|4|↔q|4(A|4|"b|4B)|4α=↓|4A|4|"b|4B. |πIdentity 
 4268  and zero laws |ε|9|1|4|"b|4A|4α=↓|4A, |9|1|4|↔q|4A|4α=↓|4A, 
 4273  |9|1|4|↔Q|4A|4α=↓|4|9|1, |πwhere |9|1 |πis the 
 4278  empty multiset. Counting law |εA|4|"b|4B|4α=↓|4(A|4|↔q|4B)|4
 4282  |"b|4(A|4|↔Q|4B). |πFurther properties analogous 
 4286  to those of sets come from the partial ordering 
 4295  de_ned by the rule |εA|4|↔Y|4B |πi= |εA|4|↔Q|4B|4α=↓|4A 
 4302  (|πi= |εA|4|↔q|4B|4α=↓|4B).|'!!|1|1|*/Notes: |π|\Other 
 4306  common applications of multisets are zeros and 
 4313  poles of meromorphic functions, invariants of 
 4319  matrices in canonical form, invariants of _nite 
 4326  Abelian groups, etc.; multisets can be useful 
 4333  in combinatorial counting arguments and in the 
 4340  development of measure theory. The terminal strings 
 4347  of a noncircular context-Free grammar form a 
 4354  multiset which is a set if and only if the grammar 
 4365  is unambigous. Although multisets appear frequently 
 4371  in mathematics, they often must be treated rather 
 4379  clumsily because there is currently no standard 
 4386  way to treat sets with repeated elements. Several 
 4394  mathemaicians have voiced their belief that the 
 4401  lack of adequate terminology and notation for 
 4408  this common concept has been a de_nite handicap 
 4416  to the development of mathematics. (A multiset 
 4423  is, of course, formally equivalent to a mapping 
 4431  from a set into the nonnegative integers, but 
 4439  this formal equivalence is of little or no practical 
 4448  value for creative mathematical reasoning.) The 
 4454  author has discussed this matter with many people 
 4462  in an attempt to _nd a good remedy. some of the 
 4473  names suggested for the concept were list, bunch, 
 4481  heap, sample, weighted set, collection; but these 
 4488  words either con⊗ict with present terminology, 
 4494  have an improper connotation, or are too much 
 4502  of a mouthful to say and to write conveniently. 
 4511  It does not seem out of place to coin a new word 
 4523  for such an important concept, and ``multiset'' 
 4530  has been suggested by N. G. de Bruijn. The notation 
 4540  ``|εA|4|"b|4B'' |πhas been selected by the author 
 4547  to avoid con⊗ict with existing notations and 
 4554  to stress the analogy with set union. It would 
 4563  not be as desirable to use ``|εA|4α+↓|4B'' |πfor 
 4571  this purpose, since algebraists have found that 
 4578  |εA|4α+↓|4B |πis a good notation for |¬T|ε|≤a|4α+↓|4|≤b|4|¬G
 4584  |4|≤a|4|¬A|4A |πand |ε|≤b|4|¬A|4B|¬Y. |πIf |εA 
 4589  |πis a multiset of nonnegative integers, let 
 4596  |εG(z)|4α=↓|4|¬K|βn|β|¬A|βA|4z|gn |πbe a generating 
 4600  function corresponding to |εA. (|πGenerating 
 4605  functions with nonnegative integer coe∃cients 
 4610  obviously correspond one-to-one with multisets 
 4615  of nonnegative integers.) If |εG(z) |πcorresponds 
 4621  to |εA |πand |εH(z) |πto |εB, |πthen |εG(z)|4α+↓|4H(z) 
 4629  |πcorresponds to |εA|4|"b|4B |πand |εG(z)H(z) 
 4634  |πcorresponds to |εA|4α+↓|4B. |πIf we form ``Drichlet'' 
 4641  generating functions |εg(z)|4α=↓|4|¬K|βn|β|¬A|βA|41/n|gz, 
 4644  h(z)|4α=↓|4|¬K|βn|β|¬A|βB|41/n|gz, |πthe product 
 4647  |εg(z)h(z) |πcorresponds to the multiset product 
 4653  |εAB.|'{A3}|π|≡2|≡0|≡.|9|4Type 3: (|εS|β0,|4.|4.|4.|4,|4S|βr
 4656  )|4α=↓|4(M|β0|β0,|4.|4.|4.|4,|4M|βr|β0)|4α=↓|4(|¬T0|¬Y,|4.|4
 4656  .|4.|4,|4|¬TA|¬Y, |¬TA|4α_↓|41,|4A|¬T, |¬TA|4α_↓|41, 
 4659  A,|4 A|¬Y, |¬TA|4α_↓|41, A|4α_↓|41, A, A, A|¬Y,|4.|4.|4.|4, 
 4666  |¬TA|4α+↓|4C|4α_↓|43, A|4α+↓|4C|4α_↓|42, A|4α+↓|4C|4α_↓|42, 
 4669  A|4α+↓|4C|4α_↓|42|¬Y). |πType 5: (|εM|β0|β0,|4.|4.|4.|4, 
 4673  M|βr|β0)|4α=↓|4(|¬T0|¬Y,|4.|4.|4.|4, |¬TA|¬Y, 
 4675  |¬TA|4α_↓|41, A|¬Y,|4.|4.|4.|4, |¬TA|4α+↓|4C|4α_↓|41, 
 4678  A|4α+↓|4C|¬Y, |¬Ta|4α+↓|4C|4α_↓|41, A|4α+↓|4C|4α_↓|41, 
 4681  A|4α+↓|4C|¬Y,|4.|4.|4.|4, |¬TA|4α+↓|4C|4α+↓|4D|4α_↓|41, 
 4683  A|4α+↓|4C|4α+↓|4D|4α_↓|41, A|4α+↓|4C|4α+↓|4D|¬Y), 
 4685  (M|β0|β1,|4.|4.|4.|4, M|βr|β1)|4α=↓|4(|9|1,|4.|4.|4.|4,|4|9|
 4686  1, |9|1,|4.|4.|4.|4,|4|9|1, |¬TA|4α+↓|4C|4α_↓|42|¬Y,|4.|4.|4
 4688  .|4, |¬TA|4α+↓|4c|4α+↓|4D|4α_↓|42|¬Y), S|βi|4α=↓|4M|βi|β0|4|
 4690  "b|4M|βi|β1.|'{A3}|π|≡2|≡1|≡.|9|4|πFor example, 
 4693  let |εu|4α=↓|42|g8|gq|gα+↓|g5,|4x|4α=↓|4(2|g(|gq|gα+↓|g1|g)|
 4694  gu|4α_↓|41)/(2|gu|4α_↓|41)|4α=↓|42|gq|gu|4α+↓|4αo↓|4αo↓|4αo↓
 4694  |4α+↓|42|gu|4α+↓|41, y|4α=↓|42|g(|gq|gα+↓|g1|g)|gu|4α+↓|41. 
 4696  |πThen |εxy|4α=↓|4(2|g2|g(|gq|gα+↓|g1|g)|gu|4α_↓|41)/(2|gu|4
 4697  α_↓|41); |πif |εn|4α=↓|42|g4|g(|gq|gα+↓|g1|g)|gu|4α+↓|4xy, 
 4700  |πwe have |λ3|ε(n)|4|¬E|44(q|4α+↓|41)u|4α+↓|4q|4α+↓|42 
 4703  |πby Theorem F, but |ε|λ3|¬⊂(n)|4α=↓|44(q|4α+↓|41)u|4α+↓|42q
 4707  |4α+↓|42 |πby Theorem H.|'{A3}|≡2|≡2|≡.|9|4Underline 
 4712  everything except the |εu|4α_↓|41 |πinsertions 
 4717  used in the calculation of |εx.|'{A3}|π|≡2|≡3|≡.|9|4Theorem 
 4724  G (everything underlined).|'{A3}|≡2|≡4|≡.|9|4Use 
 4728  the numbers (|εB|ga|ri|4α_↓|41)/(B|4α_↓|41), 
 4731  0|4|¬E|4i|4|¬E|4r, |πunderlined when |εa|βi |πis 
 4736  underlined; and |εc|βkB|gi|gα_↓|g1(B|gb|rj|4α_↓|41)/(B|4α_↓|
 4738  41) |πfor |ε0|4|¬E|4j|4|¬W|4t, 0|4|¬W|4i|4|¬E|4k|4|¬E|4|λ3|g
 4741  0(B), |πunderlined when |εc|βk |πis underlined, 
 4747  where |εc|β0,|4c|β1,|4.|4.|4. |πis a minimum 
 4752  length |λ3|g0-chain for |εB. |πTo prove the second 
 4760  inequality, let |εB|4α=↓|42|gm |πand use (3). 
 4766  (The second inequapty, is rarely, if ever, an 
 4774  improvement on Theorem G.)|'|Hu*?*?*?*?{U0}{H9L11M29}|πW58320#Co
folio 824 galley 6
 4778  mputer|4Programming!(Knuth/Addison-Wesley)!Answers!f.|4824!g
 4778  .|46d|'{A6}|≡2|≡5|≡.|9|4We may assume that |εd|βk|4α=↓|41. 
 4784  |πUse the rule |εR|4A|βk|βα_↓|β1|4.|4.|4.|4A|β1, 
 4788  |πwhere |εA|βj|4α=↓|4``XR'' |πif |εd|βj|4α=↓|41, 
 4792  A|βj|4α=↓|4|π``R'' otherwise, and where ``R'' 
 4797  means take the square root, ``X'' means multiply 
 4805  by |εx. |πFor example, if |εy|4α=↓|4(.1101101)|β2, 
 4811  |πthe rule is R R XR XR R XR XR. (There exist 
 4823  binary square-root extraction algorithms suitable 
 4828  for computer hardware, requiring an execution 
 4834  time comparable to that of division; computers 
 4841  with such hardware could also calculate more 
 4848  general fractional powers with the technique 
 4854  in this exercise.)|'{A3}|≡2|≡6|≡.|9|4If we know 
 4860  the pair (|εF|βk,|4F|βk|βα_↓|β1), |πthen (|εF|βk|βα+↓|β1,|4F
 4864  |βk)|4α=↓|4(|εF|βk|4α+↓|4F|βk|βα_↓|β1,|4F|βk) 
 4865  |πand (|εF|β2|βk,|4F|β2|βk|βα_↓|β1)|4α=↓|4(F|ur2|)k|)|4α+↓|4
 4866  2F|βkF|βk|βα_↓|β1, F|ur2|)k|)|4α+↓|4F|ur2|)kα_↓1|)); 
 4868  |πso a binary method can be used to calculate 
 4877  (|εF|βn,|4F|βn|βα_↓|β1), |πusing |εO(|πlog|4|εn) 
 4880  |πarithmetic operations. Perhaps better is to 
 4886  use the pair of values (|εF|βk,|4L|βk), |πwhere 
 4893  |εL|βk|4α=↓|4F|βk|βα_↓|β1|4α+↓|4F|βk|βα+↓|β1 
 4894  (|πcf. Section 4.5.4); then (|εF|βk|βα+↓|β1,|4L|βk|βα+↓|β1)|
 4898  4α=↓|4{H11}({H9}|f1|d32|)(F|βk|4α+↓|4L|βk), |f1|d32|)(5F|βk|
 4899  4α+↓|4L|βk){H11}){H9}, (F|β2|βk,|4L|β2|βk)|4α=↓|4(F|βkL|βk, 
 4901  L|ur2|)k|)|4α_↓|42(|→α_↓1)|gk{H11}){H9}.|'!!|1|1|πFor 
 4903  the general linear recurrence |εx|βn|4α=↓|4a|β1x|βn|βα_↓|β1|
 4907  4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|4a|βdx|βn|βα_↓|βd, |πwe 
 4909  can compute |εx|βn |πin |εO(d|g3|4|πlog|4|εn) 
 4914  |πarithmetic operations by computing the |εn|πth 
 4920  power of an appropriate |εd|4d |πmatrix. (This 
 4927  observation is due to J. C. P. Miller and D. 
 4937  J. S. Brown, |εComp. J. |π|≡9 (1966), 188<190.)|'
 4945  {A3}|≡2|≡7|≡.|9|4First form the |ε2|gm|4α_↓|4m|4α_↓|41 
 4949  |πproducts |εx|ure|β1|)1|)|4.|4.|4.|4x|ure|βm|)m|), 
 4951  |πwhere |ε0|4|¬E|4e|βj|4|¬E|41 |πand |εe|β1|4α+↓|4αo↓|4αo↓|4
 4954  αo↓|4α+↓|4e|βm|4|¬R|42. |πThen if |εn|βj|4α=↓|4(d|βj|β|≤l|4.
 4957  |4.|4.|4d|βj|β1d|βj|β0)|β2, |πthe sequence begins 
 4961  with |εx|urd|β1|≤l|)1|)|4.|4.|4.|4x|urd|βm|β|≤l|)m|) 
 4963  |πand then we square, and multiply by |εx|urd|β1|βi|)1|)|4.|
 4970  4.|4.|4x|urd|βm|βi|)m|), |πfor |εi|4α=↓|4|≤l|4α_↓|41,|4.|4.|
 4972  4.|4,|41,|40. {H11}({H9}|πStraus [|εAMM |≡7|≡1 
 4976  (1964), 807<808] |πhas shown that |ε2|≤l(n) |πmay 
 4983  be replaced by (1|4α+↓|4|ε|≤e)|≤l(n) |πfor any 
 4989  |ε|≤e|4|¬Q|40, |πby generalizing this binary 
 4994  method to |ε2|gk-|πary as in Theorem D. At least 
 5003  |λ3|ε(n|β1|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|4n|βm) |πmultiplications
 5004   are obviously required.{H11}){H9}|'{U0}{H9L11M29}|≡2|≡8|≡.|
 5008  9|4(a) |εx|4|4|4y|4α=↓|4x|4|↔I|4y|4|↔I|4(x|4α+↓|4y), 
 5010  |πwhere ``|→|↔I'' is logical ``or'', cf. exercise 
 5017  4.6.2<26; clearly |ε|≤n(x|4|4|4y)|4|¬E|4|≤n(x|4|↔I|4y)|4α+↓|
 5019  4|≤n(x|4|↔i|4y)|4α=↓|4|≤n(x)|4α+↓|4|≤n(y). |π(b) 
 5021  Note _rst that |εA|βi|βα_↓|β1/2|gd|ri|rα_↓|r1|4|↔Y|4A|βi/d|g
 5024  i |πfor |ε1|4|¬E|4r. |πSecondly, note that |εd|βj|4α=↓|4d|βi
 5030  |βα_↓|β1 |πin a nondoubling; for otherwise |εa|βi|βα_↓|β1|4|
 5036  ¬R|42a|4|¬R|4a|βj|4α+↓|4a|βk|4α=↓|4a|βi. |πHence 
 5038  |εA|βj|4|↔Y|4A|βi|βα_↓|β1 |πand |εA|βk|4|↔Y|4A|βi|βα_↓|β1/2|
 5040  gd|rj|gα_↓|gd|rk. (|πc) An easy induction on 
 5046  |εi, |πexcept that close steps need closer attention. 
 5054  Let us say that |εm |πhas property |εP(|≤a) |πif 
 5063  the 1's in its binary representation all appear 
 5071  in consecutive blocks of |ε|→|¬R|≤a |πin a row. 
 5079  If |εm |πand |εm|¬S |πhave |εP(|≤a), |πso does 
 5087  |εm|4|4|4m|¬S; |πif |εm |πhas |εP(|≤a) |πthen 
 5093  |ε|≤r(m) |πhas |εP(|≤a|4α+↓|4|≤d). |πHence |εB|βi 
 5098  |πhas |εP(1|4α+↓|4|≤dc|βi). |πFinally if |εm 
 5103  |πhas |εP(|≤a) |πthen |ε|≤n{H11}({H9}|≤r(m))|4|¬E|4(|≤a|4α+↓
 5106  |4|≤d)|≤n(m)/|≤a; |πfor |ε|≤n(m)|4α=↓|4|≤n|β1|4α+↓|4αo↓|4αo↓
 5108  |4αo↓|4α+↓|4|≤n|βq, |πwhere each block size |ε|≤n|βj 
 5114  |πis |ε|→|¬R|≤a, |πhence |ε|≤n{H11}({H9}|≤r(m){H11}){H9}|4|¬
 5117  E|4(|≤n|β1|4α+↓|4|≤d)|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|4(|≤n|βq|4α+↓
 5117  |48)|4|¬E|4(1|4α+↓|4|≤d/|≤a)|≤n|β1|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|
 5117  4(1|4α+↓|4|≤d/|≤a)|≤n|βq. |π(d) Let |εf|4α=↓|4b|βr|4α+↓|4c|β
 5120  r |πbe the number of nondoublings and |εs |πthe 
 5129  number of small steps. If |εf|4|¬R|43.271|4|πlg|4|ε|≤n(n) 
 5135  |πwe have |εs|4|¬R|4|πlg|4|ε|≤n(n) |πas desired, 
 5140  by (16). Otherwise we have |εa|βi|4|¬E|4(1|4α+↓|42|gα_↓|g|≤d
 5145  )|gb|ri2|gc|ri|gα+↓|gd|ri |πfor |ε0|4|¬E|4i|4|¬E|4r, 
 5148  |πhence |εn|4|¬E|4{H11}({H9}(1|4α+↓|42|gα_↓|g|≤d)/2{H11}){H9
 5149  }|gb|rr2|gr, |πand |εr|4|¬R|4|πlg|4|εn|4α+↓|4b|βr|4α_↓|4b|βr
 5151  |4|πlg(1|4α+↓|42|ε|gα_↓|g|≤d)|4|¬R|4|πlg|4|εn|4α+↓|4|πlg|4|ε
 5151  |≤n(n)|4α_↓|4|πlg(|ε1|4α+↓|4|≤dc|βr)|4α_↓|4b|βr|4|πlg(1|4α+↓
 5151  |42|ε|gα_↓|g|≤d). |πLet |ε|≤d|4α=↓|4|"p|πlg(|εf|4α+↓|41)|"P;
 5153   |πthen ln(|ε1|4α+↓|42|gα_↓|g|≤d)|4|¬E|4|πln{H11}({H9}1|4α+↓
 5155  |41/(|εf|4α+↓|41){H11}){H9}|4|¬E|41/(f|4α+↓|41)|4|¬E|4|≤d/(1
 5155  |4α+↓|4|≤df), |πand it follows that lg(1|4α+↓|4|ε|≤dx)|4α+↓|
 5160  4(f|4α_↓|4x)|πlg(1|4α+↓|4|ε2|gα_↓|g|≤d)|4|¬E|4|πlg(|ε1|4α+↓|
 5160  4|≤df) |πfor |ε0|4|¬E|4x|4|¬E|4f. |πHence _nally 
 5165  |ε|λ3(n)|4|¬R|4|πlg|4|εn|4α+↓|4|πlg|4|ε|≤n(n)|4α_↓|4|πlg{H11
 5165  }({H9}1|4α+↓|4(3.271|4lg|4|ε|≤n(n){H11}){H9}|"p|πlg{H11}({H9
 5165  }|ε1|4α+↓|43.271|4|πlg|4|ε|≤n(n){H11}){H11})|"P){H9}. 
 5166  |ε[Theoretical Computer Science |π|≡1 (1975), 
 5171  1<12.]|'|H|≡2|≡9|≡.|9|4In the paper just cited, 
 5177  Sch|=4onhage has re_ned the method of exercise 
 5184  28 to prove that |ε|λ3(n)|4|¬R|4|πlg|4|εn|4α+↓|4|πlg|4|ε|≤n(
 5188  n)|4α_↓|42.13 |πfor all |εn. |πCan the remaining 
 5195  gap be closed?|'{A3}{H9}|≡3|≡0|≡.|9|4|εn|4α=↓|431 
 5199  |πis the smallest example; |ε|λ3(31)|4α=↓|47, 
 5204  |πbut 1, 2, 4, 8, 16, 32, 31 |πis an addition-subtraction 
 5215  chain of length 6. Erd|=4os has stated that Theorem 
 5224  E holds also for addition-subtraction chains. 
 5230  Sch|=4onhage has extended his lower bound of 
 5237  exercise 28 to addition-subtraction chains, with 
 5243  |ε|≤n(n) |πreplaced by |ε|=3|≤n(n)|4α=↓|4|πminimum 
 5247  number of nonzero digits to represent |εn|4α=↓|4(n|βq|4.|4.|
 5253  4.|4n|β0)|β2 |πwhere each |εn|βj |πis |→α_↓1, 
 5259  0, or |→α+↓1; |ε|=3|≤n(n) |πis the number of 
 5267  1's, in the ordinary binary representation of 
 5274  |εn, |πwhich are immediately preceded by 0 |πor 
 5282  by the string 00(10)|ε|gk1 |πfor some |εk|4|¬R|40.)|'
 5289  {A3}|π|≡3|≡2|≡.|9|4Andrew C. Yao has essentially 
 5294  found the asymptotic behavior for _xed |εm, |πby 
 5302  generalizing the 2|ε|gk-|πary method to obtain 
 5308  the upper bound |ε|≤l(n|βm)|4α+↓|4O{H11}({H9}|¬K|ur|)1|¬Ei|¬
 5311  Em|)|4|≤l(n|βi)/|≤l|≤l(n|βi){H11}){H9}.|'{A24}{H10}|π|∨S|∨E|
 5312  ∨C|∨T|∨I|∨O|∨N |∨4|∨.|∨6|∨.|∨4|'{A12}{H9}|9|1|≡1|≡.|9|4Set 
 5315  |εy|4|¬L|4x|g2, |πthen compute {H11}({H9}(αo↓|4αo↓|4αo↓|4(|ε
 5318  u|β2|βn|βα+↓|β1y|4α+↓|4u|β2|βn|βα_↓|β1)y|4α+↓|4u|β1{H11}){H9
 5318  }x.|'{A3}|9|1|≡2|≡.|9|4|πReplacing |εx |πin (2) 
 5323  |πby the polynomial |εx|4α+↓|4x|β0 |πleads to 
 5329  the following procedure:|'{A3}{I3.1H}!!|1|1|≡G|≡1|≡.|9Do 
 5333  step G2, for |εk|4α=↓|4n, n|4α_↓|41,|4.|4.|4.|4,|40 
 5338  (|πin this order), and stop.|'{A3}!!|1|1|≡G|≡2|≡.|9Set 
 5344  |εv|βk|4|¬L|4u|βk, |πand then set |εv|βj|4|¬L|4v|βj|4α+↓|4x|
 5348  β0v|βj|βα+↓|β1 |πfor |εj|4α=↓|4k, k|4α+↓|41,|4.|4.|4.|4, 
 5352  n|4α_↓|41. (|πWhen |εk|4α=↓|4n, |πthis step simply 
 5358  sets |εv|βn|4|¬L|4u|βn.)|'{A3}{IC}|πThe computations 
 5362  turn out to be identical to those in H1, H2, 
 5372  but performed in a di=erent order.|'{A3}|9|1|≡3|≡.|9|4The 
 5379  coe∃cient of |εx|gk |πis a polynomial in |εy 
 5387  |πwhich may be evaluated by Horner's rule: {H11}({H9}αo↓|4αo
 5394  ↓|4αo↓|4(|εu|βn|β,|β0x|4α+↓|4(u|βn|βα_↓|β1|β,|β1y|4α+↓|4u|βn
 5394  |βα_↓|β1|β,|β0))x|4α+↓|4αo↓|4αo↓|4αo↓{H11}){H9}x|4α+↓|4{H11}
 5394  ({H9}(αo↓|4αo↓|4αo↓|4(u|β0|β,|βny|4α+↓|4u|β0|β,|βn|βα_↓|β1)y
 5394  |4α+↓|4αo↓|4αo↓|4αo↓)y|4α+↓|4u|β0|β,|β0{H11}){H9}. 
 5395  (|πFor a ``homogeneous'' polynomial, such as 
 5401  |εu|βnx|gn|4α+↓|4u|βn|βα_↓|β1x|gn|gα_↓|g1y|4α+↓|4αo↓|4αo↓|4α
 5401  o↓|4α+↓|4u|β1xy|gn|gα_↓|g1|4α+↓|4u|β0y|gn, |πanother 
 5403  scheme is more e∃cient: _rst divide |εx |πby 
 5411  |εy, |πevaluate a polynomial in |εx/y, |πthen 
 5418  multiply by |εy|gn.{H11}){H9}|'{A3}|9|1|≡4|≡.|9|4|πRule 
 5422  (2) involves |ε4n |πor |ε3n |πreal multiplications 
 5429  and |ε4n |πor |ε7n |πreal additions; (3) is worse, 
 5438  it takes 4|εn|4α+↓|42 |πor |ε4n|4α+↓|41 |πmults, 
 5444  |ε4n|4α+↓|42 |πor |ε4n|4α+↓|45 |πodds.|'|π|9|1|≡5|≡.|9|1One 
 5449  multiplication to compute |εx|g2; |"ln/2|"L |πmultiplication
 5454  s and |"l|εn/2|"L |πadditions to evaluate the 
 5461  _rst line; |"p|εn/2|"P |πmultiplications and 
 5466  |ε|"pn/2|"P|4α_↓|41 |πadditions to evaluate the 
 5471  second line; and one addition to add the two 
 5480  lines together. Total: |εn|4α+↓|41 |πmultiplications 
 5485  and |εn |πadditions.|'{A3}|9|1|≡6|≡.|9|4|≡J|≡1|≡.|9Compute 
 5489  and store the values |εx|ur2|)0|),|4.|4.|4.|4,|4x|urjα_↓|"ln
 5493  /2|"L|)0|).|π|'{A3}!!|1|1|π|≡J|≡2|≡.|9|4Set |εv|βj|4|¬L|4u|β
 5495  jx|urjα_↓|"ln/2|"L|)0|) |πfor |ε0|4|¬E|4j|4|¬E|4n.|'
 5498  {A3}!!|1|1|π|≡J|≡3|≡.|9For |εk|4α=↓|40,|41,|4.|4.|4.|4,|4n|4
 5499  α_↓|41, |πset |εv|βj|4|¬L|4v|βj|4α+↓|4v|βj|βα+↓|β1 
 5502  |πfor |εj|4α=↓|4n|4α_↓|41,|4.|4.|4.|4,|4k|4α+↓|41,|4k.|'
 5504  {A3}!!|1|1|π|≡J|≡4|≡.|9Set |εv|βj|4|¬L|4v|βjx|ur|"ln/2|"Lα_↓
 5505  j|)0|) |πfor |ε0|4|¬E|4j|4|¬E|4n.|'{A3}|πThere 
 5509  are (|εn|4α+↓|4n|g2)/2 |πadditions, |εn|4α+↓|4|"pn/2|"P|4α_↓
 5512  |41 |πmultiplications, |εn |πdivisions. Another 
 5517  multiplication and division can be saved by treating 
 5525  |εv|βn |πand |εv|β0 |πas special cases. |εReference|*/:|\ 
 5532  SIGACT News |≡7|≡, |π3 (Summer 1975), 32<34.|'
 5539  {A3}|9|1|≡7|≡.|9|1|3Let |εx|βj|4α=↓|4x|β0|4α+↓|4jh, 
 5541  |πand consider (42), (44). Set |εy|βj|4|¬L|4u(x|βj) 
 5547  |πfor |ε0|4|¬E|4j|4|¬E|4n. |πFor |εk|4α=↓|41, 
 5551  2,|4.|4.|4.|4,|4n |π(in this order), set |εy|βj|4|¬L|4y|βj|4
 5556  α_↓|4y|βj|βα_↓|β1 |πfor |εj|4α=↓|4k,|4k|4α+↓|41,|4.|4.|4.|4,
 5558  |4n (|πin this order). Now |ε|≤b|βj|4α=↓|4y|βj 
 5564  |πfor all |εj.|'{A3}|9|1|π|≡8|≡.|9|4See (43).|'
 5569  {A3}|9|1|≡9|≡.|9|4[|εCombinatorial Mathematics 
 5571  (|πBu=alo: Math. Ass'n of America, 1963), 26<28.] 
 5578  This formula can be regarded as an application 
 5586  of the principle of inclusion and exclusion (Section 
 5594  1.3.3), since the sum of the terms for |εn|4α_↓|4|≤e|β1|4α_↓
 5602  |4αo↓|4αo↓|4αo↓|4α_↓|4|≤e|βn|4α=↓|4k |πis the 
 5605  sum of all |εx|β1|βj|m1x|β2|βj|m2|4αo↓|4αo↓|4αo↓|4x|βn|βj|mn
 5608   |πfor which |εk |πvalues of the |εj, |πdo not 
 5618  appea. A direct proof can be given by observing 
 5627  that the coe∃cient of |εx|β1|βj|m1|4.|4.|4.|4x|βn|βj|mn 
 5632  |πis|'{A9}|ε|¬K|4(|→α_↓1)|gn|gα_↓|g|≤e|r1|gα_↓|1|gαo↓|1|gαo↓
 5633  |1|gαo↓|1|gα_↓|g|≤e|rn|≤e|βj|m1|4.|4.|4.|4|≤e|βj|mn;|;
 5634  {A9}|πif the |εj|π's are distinct, this equals 
 5641  unity, but if |εj|β1,|4.|4.|4.|4,|4j|βn|4|=|↔6α=↓|4k 
 5645  |πthen it is zero, since the terms for |ε|≤e|βk|4α=↓|40 
 5654  |πcancel the terms for |ε|≤e|βk|4α=↓|41.|'!!|1|1|πTo 
 5660  evaluate the sum e∃ciently, we can start with 
 5668  |ε|≤e|β1|4α=↓|41, |≤e|β2|4α=↓|4αo↓|4αo↓|4αo↓|4α=↓|4|≤e|βn|4α
 5669  =↓|40, |πand we can then proceed through all 
 5677  combinations of the |ε|≤w|π's in such a way that 
 5686  only one |ε|≤e |πchanges from one term to the 
 5695  ne t. (See ``Gray code'' in Chapter 7.) The work 
 5705  to compute the _rst term is |εn|4α_↓|41 |πmultiplications; 
 5713  the subsequent 2|g|εn|4α_↓|42 |πterms each involve 
 5719  |εn |πadditions, then |εn|4α_↓|41 |πmultiplications, 
 5724  then one more addition. Total: (|ε2|gn|4α_↓|41)(n|4α_↓|41) 
 5730  |πmultiplications, and (|ε2|gn|4α_↓|42)(n|4α+↓|41) 
 5733  |πadditions. Only |εn|4α+↓|41 |πtemp storage 
 5738  locations are needed, one for the main partial 
 5746  sum and one for each factor of the cur{U0}{H9L11M29}|πW58320
folio 828 galley 7
 5754  #Computer|4Programming!(Knuth/Addison-Wesley)!Answers!f.|482
 5754  8!g.|47d|'{A6}|ε|≡1|≡0|≡.|9|4|ε|¬K|β1|β|¬E|βk|β|¬W|βn|4(k|4α
 5755  +↓|41)(|fn|d5kα+↓1|))|4α=↓|4n(2|gn|gα_↓|g1|4α_↓|41) 
 5756  |πmultiplications and |ε|¬K|β1|β|¬E|βk|β|¬W|βn|4k(|fn|d5kα+↓
 5758  1|))|4α=↓>n2|gn|gα_↓|g1|4α_↓|42|gn|4α+↓|41 |πadditions. 
 5761  This is approximately half as many arithmetic 
 5768  operations as the method of exercise 9, although 
 5776  it requires a more complicated program to control 
 5784  the sequence. Approximately (|ε|fn|d5|"pn/2|"P|))|4α+↓|4(|fn
 5787  |d5|"pn/2|"Pα_↓1|)) |πtemporary storage locations 
 5791  must be used, and this grows exponentially large 
 5799  (on the order of 2|ε|gn/|¬H|v4n|)4|gn/n|g1|g.|g5.|'
 5804  !!|1|1|πThe method in this exercise is equivalent 
 5811  to the unusual matrix factorization of the permanent 
 5819  function, given by Jurkat and Ryser in |εJ. Algebra 
 5828  |≡5 (|π1967), 342<357. It may also be regarded 
 5836  as an application of (39), (40), in an appropriate 
 5845  sense.|'{A3}|≡1|≡2|≡.|9|4(J. Hopcroft and L. 
 5850  R. Kerr have shown among other things that 7 
 5859  multiplications are necessary in 2|4α⊗↓|42 matrix 
 5865  multiplication [|εSIAM J. Appl. Math. |π|≡2|≡0 
 5871  (1971), 30<36]. R. L. Probert has shown that 
 5879  all 7-multiplication schemes, in which each multiplication 
 5886  takes a linear combination of elements from one 
 5894  matrix and multiplies by a linear combination 
 5901  of elements from the other, must have at least 
 5910  15 additions [|εSIAM J. Computing, |πto appear]. 
 5917  The best lower bound now known for large |εn 
 5926  |πis |ε2n|g2|4α_↓|4n |πmultiplications, a result 
 5931  due to D. Kirkpatrick. For |εn|4α=↓|43, |πJ. 
 5938  D. Laderman [|εBull. Amer. Math. Soc. |π|≡8|≡2 
 5945  (1976), 126<128] has shown that 23 noncommutative 
 5952  multiplications su∃ce.)|'{A3}|≡1|≡3|≡.|9|4|εF(t|β1,|4.|4.|4.
 5954  |4,|4t|βn)|4α=↓|4{H11}({H9}|¬K|ur|)0|¬Es|β1|¬Wm|β1,|1.|1.|1.
 5954  |1,|10|¬Es|βn|¬Wm|βn|)|4|πexp{H11}({H9}|ε|→α_↓2|≤pi(s|β1t|β1
 5954  /m|β1|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|4s|βnt|βn/m|βn)f(s|β1,|4.|4.|
 5954  4.|4,|4s|βn){H11}){H9}/m|β1,|4.|4.|4.|4m|βn, 
 5955  |πby summing geometric series. The inverse transform 
 5962  times |εm|β1|4.|4.|4.|4m|βn |πcan be found by 
 5968  doing a regular transform and interchanging |εt|βj 
 5975  |πwith |εm|βj|4α_↓|4t|βj |πwhen |εt|βj|4|=|↔6α=↓|40, 
 5979  |πcf. exercise 4.3.3<15.|'!!|1|1{H11}(If we regard 
 5985  |εF(t|β1,|4.|4.|4.|4,|4t|βn) |πas the coe∃cient 
 5989  of |εx|urt|β1|)1|)|4.|4.|4.|4x|urt|βn|)n|) |πin 
 5992  a multivariate polynomial, the _nite Fourier 
 5998  transform amounts to evaluation of this polynomial 
 6005  at roots of unity, and the inverse transform 
 6013  amounts to _nding the interpolating polynomial.{H11})|'
 6019  {A3}|≡1|≡4|≡.|9|4(a) Let |εm|4α=↓|42, F(t|β1,|4t|β2,|4.|4.|4
 6022  .|4,|4t|βn)|4α=↓|4F(2|gn|gα_↓|g1t|βn|4α+↓|4αo↓|4αo↓|4αo↓|4α+
 6022  ↓|42t|β2|4α+↓|4t|β1), f(s|β1,|4s|β2,|4.|4.|4.|4,|4s|βn)|4α=↓
 6023  |4f(2|gn|gα_↓|g1s|β1|4α+↓|42|gn|gα_↓|g2s|β2|4α+↓|4αo↓|4αo↓|4
 6023  αo↓|4α+↓|4s|βn); |πnote the reversed treatment 
 6028  between |εt|π's and |εs|π's. Also let |εg|βk(s|βk,|4.|4.|4.|
 6034  4,|4s|βn,|4t|βk) |πbe |ε|≤v |πraised to the 2|ε|gk|gα_↓|g1t|
 6040  βk(s|βn|4α+↓|42s|βn|βα_↓|β1|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|42|gn|g
 6040  α_↓|gks|βk) |πpower. It is possible to gain some 
 6048  speed by combining steps |εk |πand |εk|4α+↓|41, 
 6055  |πfor |εk|4α=↓|41,|43,|4.|4.|4.; |πthis expedites 
 6059  several of the computations of sines and cosines. 
 6067  [The fast Fourier transform algorithm is essentially 
 6074  due to C. Runge and H. K|=4onig in 1924, and 
 6084  it was generalized by J. W. Cooley and J. W. 
 6094  Tukey, |εMath. Comp. |π|≡1|≡9 (1965), 297<301. 
 6100  Its interesting history has been traced by J.uW. 
 6108  Cooley, P. A. W. Lewis, P. D. Welch, |εProc. 
 6117  IEEE |π|≡5|≡5 (1967), 1675<1677. Details concerning 
 6123  its use have been discussed by R. C. Singleton, 
 6132  |εC|4ACM |≡1|≡0 (1967), |π647<654; M. C. Pease, 
 6139  |εJ|4ACM |≡1|≡5 (1968), 252<264; |πG. D. Berglund, 
 6146  |εMath. Comp. |≡2|≡2 (1968), 275<279, C|4ACM 
 6152  |≡1|≡1 (1968), |π703<710.]|'!!|1|1(a) Let |εN 
 6158  |πbe the smallest power of 2 which exceeds |ε2n, 
 6167  |πand let |εu|βn|βα+↓|β1|4α=↓|4αo↓|4αo↓|4αo↓|4α=↓|4u|βN|βα_↓
 6169  |β1|4α=↓|4v|βn|βα+↓|β1|4α=↓|4αo↓|4αo↓|4αo↓|4α=↓|4v|βN|βα_↓|β
 6169  1|4α=↓|40. |πIf |εU|βs|4α=↓|4|¬K|ur|)0|¬Et|¬WN|)|4u|βt|≤v|gs
 6171  |gt, V|βs|4α=↓|4|¬K|ur|)0|¬Et|¬WN|)|4v|βt|≤v|gs|gt, 
 6173  0|4|¬E|4s|4|¬W|4N, |≤v|4α=↓|4e|g2|g|≤p|gi|g/|gN, 
 6175  |πthen |ε|¬K|ur|)0|¬Es|¬WN|)|4U|βsV|βs|≤v|gα_↓|gs|gt|4α=↓|4N
 6176  |4|¬K|4u|βt|m1v|βt|m2. |πThe latter sum is taken 
 6182  over all |εt|β1 |πand |εt|β2 |πwith |ε0|4|¬E|4t|β1, 
 6189  t|β2|4|¬W|4N, t|β1|4α+↓|4t|β2|4|"o|4t (|πmodulo 
 6192  |εN). |πThe terms vanish unless |εt|β1|4|¬E|4n, 
 6198  t|β2|4|¬E|4n, |πso |εt|β1|4α+↓|4t|β2|4|¬W|4N; 
 6201  |πthus the sum is the coe∃cient of |εz|gt |πin 
 6210  the product |εu(z)v(z). |πIf we use the method 
 6218  of (a) to compute the Fourier transforms and 
 6226  the inverse transforms, the number of complex 
 6233  operations is |εO(N|4|πlog|4|εN)|4α+↓|4O(N|4|πlog|4|εN)|4α+↓
 6235  |4O(N)|4α+↓|4O(N|4|πlog|4|εN); |πand |εN|4|¬W|44n. 
 6238  [|πCf. Section 4.3.3 and paper by J. M. Pollard, 
 6247  |εMath. comp. |π|≡2|≡5 (1971), 365<374.]|'{U0}{H9L11M29}!!|1
 6252  |1The number |ε|≤v |πcannot be represented exactly 
 6259  inside a computer, but V. Strassen has shown 
 6267  that it isn't necessary to have too much accuracy 
 6276  to deduce exact results when the coe∃cients are 
 6284  integers [|εComputing |π|≡7 (1971), 281<292].|'
 6289  !!|1|1It is possible to use an |εinteger |πnumber 
 6297  |ε|≤v |πwhich is of order |ε2|gt |πmodulo a prime 
 6306  |εp, |πand to determine the results modulo su∃ciently 
 6314  many primes. Useful primes in this regard, together 
 6322  with their least primitive roots |εr (|πfrom 
 6329  which we take |ε|≤v|4α=↓|4r|g(|gp|gα_↓|g1|g)|g/|g2|it|4|πmod
 6332  |4|εp |πwhen |εp|4|πmod|4|ε2|gt|4α=↓|41), |πcan 
 6336  be found as described in Section 4.5.4. For |εt|4α=↓|49, 
 6345  |πthe largest cases |→|¬w2|g3|g5 |πare |εp|4α=↓|42|g3|g5|4α_
 6350  ↓|4512a|4α+↓|41, |πwhere (|εa,|4r)|4α=↓|4(28,|47), 
 6353  (31,|410), 34,|413), (56,|43), (58,|410), (76,|45), 
 6358  (80,|43), (85,|411), 91,|45), (101,|43); |πthe 
 6363  ten largest cases |→|¬W2|g3|g1 are |εp|4α=↓|42|g3|g1|4α_↓|45
 6368  12a|4α+↓|41, |πwhere (|εa,|4r)|4α=↓|4(1,|410), 
 6371  (11,|43), (19,|411), (20,|43), (29,|43), (35,|43), 
 6376  (55,|419), (65,|46), (95,|43), (121,|410). |πFor 
 6381  larger |εt, |πall primes |εp |πof the form |ε2|gtq|4α+↓|41 
 6390  |πwhere |εq|4|¬W|432 |πis odd and 2|g2|g4|4|¬W|4|εp|4|¬W|42|
 6395  g3|g6 |πare given by (|εp|4α_↓|41,|4r)|4α=↓|4(11|4αo↓|42|g2|
 6399  g1,|43), (25|4αo↓|42|g2|g0,|43), (27|4αo↓|42|g2|g0,|45), 
 6402  (25|4αo↓|42|g2|g2,|43), (27|4αo↓|42|g2|g2,|47), 
 6404  (5|4αo↓|42|g2|g5,|43), (7|4αo↓|42|g2|g5,|43), 
 6406  (7|4αo↓|42|g2|g6,|43), (27|4αo↓|42|g2|g6,|413), 
 6408  (15|4αo↓|42|g2|g7,|431), (17|4αo↓|42|g2|g7,|43), 
 6410  (3|4αo↓|42|g3|g0,|45), (13|4αo↓|42,|4|g2|g8,|43), 
 6412  (29|4αo↓|42|g2|g7,|43), (23|4αo↓|42|g2|g9,|45). 
 6414  |πSome of the latter primes can be used with 
 6423  |ε|≤v|4α=↓|42|ge |πfor appropriate small |εe.|'
 6428  {A3}|π|≡1|≡5|≡.|9|4(a) The hint follows by integration 
 6434  and induction. Let |εf|g(|gn|g)(|≤u) |πtake on 
 6440  all values between |εA |πand |εB |πinclusive, 
 6447  as |ε|≤u |πvaries from min(|εx|β0,|4.|4.|4.|4,|4x|βn) 
 6452  |πto max(|εx|β0,|4.|4.|4.|4,|4x|βn). |πReplacing 
 6455  |εf|g(|gn|g) |πby these bounds, in the stated 
 6462  integral, yields |εA/n*3|4|¬E|4f(x|β0,|4.|4.|4.|4,|4x|βn)|4|¬
 6464  E|4B/n*3. |π(b) It su∃ces to prove this for |εj|4α=↓|4n. 
 6473  |πLet |εf |πbe Newton's interpolation polynomial, 
 6479  then |εF|g(|gn|g) |πis the constant |εn*3|4|≤a|βn.|'
 6485  {A3}|π|≡1|≡6|≡.|9|4Carry out the multiplications 
 6489  and additions of (43) as operations on polynomials. 
 6497  (The special case |εx|β0|4α=↓|4x|β1|4α=↓|4αo↓|4αo↓|4αo↓|4α=↓
 6500  |4x|βn |πis considered in exercise 2. We have 
 6508  used this method in step C8 of Algorithm 4.3.3C.)|'
 6517  {A3}|≡1|≡7|≡.|9|4T. M. Vari has shown that |εn|4α_↓|41 
 6524  |πmultiplications are necessary, by proving that 
 6530  |εn |πare necessary to compute |εx|ur2|)1|)|4α+↓|4αo↓|4αo↓|4
 6535  αo↓|4α+↓|4x|ur2|)n|) [|πCornell Computer Science 
 6539  report 120 (Jan. 1972)].|'{A3}|≡1|≡8|≡.|9|4|ε|≤a|β0|4α=↓|4|f
 6543  1|d32|)(u|β3/u|β4|4α+↓|41), |≤b|4α=↓|4u|β2/u|β4|4α_↓|4|≤a|β0
 6544  (|≤a|β0|4α_↓|41), |≤a|β1|4α=↓|4|≤a|β0|≤b|4α_↓|4u|β1/u|β4, 
 6546  |≤a|β2|4α=↓|4|≤b|4α_↓|42|≤a|β1, |≤a|β3|4α=↓|4u|β0/u|β4|4α_↓|
 6547  4|≤a|β1(|≤a|β1|4α+↓|4|≤a|β2), |≤a|β4|4α=↓|4u|β4.|'
 6549  {A3}|π|≡1|≡9|≡.|9|4Since |ε|≤a|β5 |πis the leading 
 6554  coe∃cient, we may assume without loss of generality 
 6562  that |εu(x) |πis monic (i.e., |εu|β5|4α=↓|41). 
 6568  |πThen |ε|≤a|β0 |πis a root of the cubic equation 
 6577  40|εz|g3|4α_↓|424u|β4z|g2|4α+↓|4(4u|ur2|)4|)|4α+↓|42u|β3)z|4
 6577  α+↓|4(u|β2|4α_↓|4u|β3u|β4)|4α=↓|40; |πthis equation 
 6580  always has at least one real root, and it may 
 6590  have three. Once |ε|≤a|β0 |πis determined, we 
 6597  have |ε|≤a|β3|4α=↓|4u|β4|4α_↓|44|≤a|β0, |≤a|β1|4α=↓|4u|β3|4α
 6599  _↓|44|≤a|β0|≤a|β3|4α_↓|46|≤a|ur2|)0|), |≤a|β2|4α=↓|4u|β1|4α_
 6600  ↓|4|≤a|β0(|≤a|β0|≤a|β1|4α+↓|44|≤a|ur2|)0|)|≤a|β3|4α+↓|42|≤a|
 6600  β1|≤a|β3|4α+↓|4|≤a|ur3|)0|)), |≤a|β4|4α=↓|4u|β0|4α_↓|4|≤a|β3
 6601  (|≤a|ur4|)0|)|4α+↓|4|≤a|β1|≤a|ur2|)0|)|4α+↓|4|≤a|β2).|'
 6602  !!|1|1|πFor the given polynomial we are to solve 
 6610  the cubic equation 40|εz|g3|4α_↓|4120z|g2|4α+↓|480z|4α=↓|40;
 6613   |πthis leads to three solutions (|ε|≤a|β0, |≤a|β1, 
 6621  |≤a|β2, |≤a|β3, |≤a|β4, |≤a|β5)|4α=↓|4(0, |→α_↓10, 
 6626  13, 5, |→α_↓5, 1), (1, |→α_↓20, 68, 1, 11, 1), 
 6636  (2, |→α_↓10, 13, |→α_↓3, 27, 1).|'{A9}|≡2|≡0|≡.|9|4|∂|¬l|¬d|
 6642  ¬a|4|4|4!|∂|¬x|h|≤a|β0!|2|≤a|β3|9|7!!!!!!!!!|9|n|∂|¬f|¬a|¬d!
 6642  |∂|≤*/|≤a|β1|≤*/!!|9|'|L|¬f|¬a|¬d|¬d|L|≤$|≤a|β3|≤$|L|¬f|¬m|¬u|
 6643  ¬l|L|¬t|¬e|¬m|¬p|¬2>|L|¬s|¬t|¬a|L|¬t|¬e|¬m|¬p|¬1|L|¬f|¬a|¬d|
 6644  ¬d|L|≤$|≤a|β2|≤$>|L|¬f|¬a|¬d|¬d|L|≤$|≤a|β0|≤*/|≤a|β3|≤$|L|≤f|
 6645  ≤m|≤u|≤l|L|≤t|≤e|≤m|≤p|≤1>|L|¬s|¬t|¬a|L|¬t|¬e|¬m|¬p|¬2|L|¬f|
 6646  ¬a|¬d|¬d|L|≤$|≤a|β4|≤$>|L|¬f|¬m|¬u|¬l|L|¬t|¬e|¬m|¬p|¬2|L|¬f|
 6647  ¬m|¬u|¬l|L|≤$|≤a|β5|≤$>|L|¬s|¬t|¬a|L|¬e|¬e|¬m|¬p|¬2|'
 6649  >{A3}|π|≡2|≡1|≡.|9|4|εz|4α=↓|4(x|4α+↓|41)x|4α_↓|42, 
 6651  w|4α=↓|4(x|4α+↓|45)z|4α+↓|49, u(x)|4α=↓|4(w|4α+↓|4z|4α_↓|48)
 6652  w|4α_↓|48; |πor |εz|4α=↓|4(x|4α+↓|49)x|4α+↓|426, 
 6655  w|4α=↓|4(x|4α_↓|43)z|4α+↓|473, u(x)|4α=↓|4(w|4α+↓|4z|4α_↓|42
 6656  4)w|4α_↓|412.|'{A3}|≡2|≡2|≡.|9|4|≤a|β6|4α=↓|41, 
 6658  |≤a|β0|4α=↓|4|→α_↓1, |≤a|β1|4α=↓|41, |≤b|β1|4α=↓|4|→α_↓2, 
 6661  |≤b|β2|4α=↓|4|→α_↓2, |≤b|β4|4α=↓|41, |≤a|β3|4α=↓|4|→α_↓4, 
 6664  |≤a|β2|4α=↓|40, |≤a|β4|4α=↓|44, |≤a|β5|4α=↓|4|→α_↓2. 
 6667  |πWe form |εz|4α=↓|4(x|4α_↓|41)x|4α+↓|41, w|4α=↓|4z|4α+↓|4x,
 6670   |πand |εu(x)|4α=↓|4{H11}({H9}(z|4α_↓|4x|4α_↓|44)w|4α+↓|44{H
 6672  11}){H9}z|4α_↓|4x.|'{A3}|π|≡2|≡3|≡.|9|4a) We 
 6675  may use induction on |εn; |πthe result is trivial 
 6684  if |εn|4|¬W|42. |πIf |εf(0)|4α=↓|40, |πthen the 
 6690  result is true for the polynomial |εf(z)/z, |πso 
 6698  it holds for |εf(z). |πIf |εf(iy)|4α=↓|40 |πfor 
 6705  some real |εy|4|=|↔6α=↓|40, |πthen |εg(|→|¬Niy)|4α=↓|4h(|→|¬
 6709  Niy)|4α=↓|40; |πsince the result is true for 
 6716  |εf(z)/(z|g2|4α+↓|4y|g2), |πit holds also for 
 6721  |εf(z). |πTherefore we may assume that |εf(z) 
 6728  |πhas no roots whose real part is zero. Now the 
 6738  net number of times the given path circles the 
 6747  origin is the number of roots of |εf(z) |πinside 
 6756  the region, which is at most 1. When |εR |πis 
 6766  large, the path |εf(Re|gi|gt) |πfor |ε|≤p/2|4|¬E|4t|4|¬E|43|
 6771  ≤p/2 |πwill circle the origin clockwise approximately 
 6778  |εn/2 |πtimes; so the path |εf(it) |πfor |ε|→α_↓R|4|¬E|4t|4|
 6785  ¬E|4R |πmust go counterclockwise around the origin 
 6792  at least |εn/2|4α_↓|41 |πtimes. For |εn |πeven, 
 6799  this implies that |εf(it) |πcrosses the imaginary 
 6806  axis at least |εn|4α_↓|42 |πtimes, and the real 
 6814  axis at least |εn|4α_↓|43 |πtimes; for |εn |πodd, 
 6822  |εf(it) |πcrosses the real axis at least |εn|4α_↓|42 
 6830  |πtimes and the imganary axis at least |εn|4α_↓|43 
 6838  |πtimes. These are roots respectively of |εg(it)|4α=↓|40, 
 6845  h(it)|4α=↓|40.|'!!|1|1|πb)|9If not, |εg |πor 
 6850  |εh |πwould have a root of the form |εa|4α+↓|4bi 
 6859  |πwith |εa|4|=|↔6α=↓|40 |πand |εb|4|=|↔6α=↓|40. 
 6863  |πBut this would imply the existence of at least 
 6872  three other such roots, namely |εa|4α_↓|4bi |πand 
 6879  |→α_↓|εa|4|¬N|4bi, |πwhile |εg(z), h(z) |πhave 
 6884  at most |εn |πroots.|'{A3}|H=*?*?*?{U0}{H9L11M29}|πW58320#Compu
folio 832 galley 8
 6888  ter|4Programming!(Knuth/Addison-Wesley)!Answers!f.|4832!g.|4
 6888  8d|'{A6}|≡2|≡4|≡.|9|4The roots of |εu |πare |→α_↓7, 
 6895  |→α_↓3|4|¬N|4|εi, |→α_↓2|4|¬N|4i, |→α_↓1; |πpermissible 
 6899  values of |εc |πare |ε2 |πand 4 (|εnot |π3, since 
 6909  |εc|4α=↓|43 |πmakes the sum of the roots equal 
 6917  to zero). Case 1, |εc|4α=↓|42: p(x)|4α=↓|4(x|4α+↓|45)(x|g2|4
 6922  α+↓|42x|4α+↓|42)(x|g2|4α+↓|41)(x|4α_↓|41)|4α=↓|4x|g6|4α+↓|46
 6922  x|g5|4α+↓|46x|g4|4α+↓|44x|g3|4α_↓|45x|g2|4α_↓|42x|4α_↓|410; 
 6923  q(x)|4α=↓|46x|g2|4α+↓|44x|4α_↓|42|4α=↓|46(x|4α+↓|41)(x|4α_↓|
 6923  4|f1|d33|)). |πLet |ε|≤a|β2|4α=↓|4|→α_↓1, |≤a|β1|4α=↓|4|f1|d
 6926  33|); p|β1(x)|4α=↓|4x|g4|4α+↓|46x|g3|4α+↓|45x|g2|4α_↓|42x|4α
 6927  _↓|410|4α=↓|4(x|g2|4α+↓|46x|4α+↓|4|f16|d33|))(x|g2|4α_↓|4|f1
 6927  |d33|))|4α_↓|4|f74|d39|); |≤a|β0|4α=↓|46, |≤b|β0|4α=↓|4|f16|
 6929  d33|), |≤b|β1|4α=↓|4|→α_↓|f74|d39|). |πCase 2, 
 6933  |εc|4α=↓|44: |πA similar analysis gives |ε|≤a|β2|4α=↓|49, 
 6939  |≤a|β1|4α=↓|4|→α_↓3, |≤a|β0|4α=↓|4|→α_↓6, |≤b|β0|4α=↓|412, 
 6942  |≤b|β1|4α=↓|4|→α_↓26.|'{A3}|π|≡2|≡5|≡.|9|4|ε|≤b|β1|4α=↓|4|≤a
 6943  |β2, |≤b|β2|4α=↓|42|≤a|β1, |≤b|β3|4α=↓|4|≤a|β7, 
 6946  |≤b|β4|4α=↓|4|≤a|β6, |≤b|β5|4α=↓|4|≤b|β6|4α=↓|40, 
 6948  |≤b|β7|4α=↓|4|≤a|β1, |≤b|β8|4α=↓|40, |≤b|β9|4α=↓|42|≤a|β1|4α
 6950  _↓|4|≤a|β8.|'{A3}|π|≡2|≡6≡.|9|4(a) |ε|≤l|β1|4α=↓|4|≤a|β1|4α⊗
 6952  ↓|4|≤l|β0, |≤l|β2|4α=↓|4|≤a|β2|4α+↓|4|≤l|β1, 
 6954  |≤l|β3|4α=↓|4|≤l|β2|4α⊗↓|4|≤l|β0, |≤l|β4|4α=↓|4|≤a|β3|4α+↓|4
 6955  |≤l|β3, |≤l|β5|4α=↓|4|≤l|β4|4α⊗↓|4|≤l|β0, |≤l|β6|4α=↓|4|≤a|β
 6957  4|4α+↓|4|≤l|β5. |π(b) |ε|≤k|β1|4α=↓|41|4α+↓|4|≤b|β1x, 
 6960  |≤k|β2|4α=↓|41|4α+↓|4|≤b|β2|≤k|β1x, |≤k|β3|4α=↓|41|4α+↓|4|≤b
 6961  |β3|≤k|β2x, u(x)|4α=↓|4|≤b|β4|≤k|β3|4α=↓|4|≤b|β1|≤b|β2|≤b|β3
 6962  |≤b|β4x|g3|4α+↓|4|≤b|β2|≤b|β3|≤b|β4x|g2|4α+↓|4|≤b|β3|≤b|β4x|
 6962  4α+↓|4|≤b|β4. |π(c) If any coe∃cient is zero, 
 6969  the coe∃cient of |εx|g3 |πmust also be zero in 
 6978  (b), while (a) yields an arbitrary polynomial 
 6985  |ε|≤a|β1x|g3|4α+↓|4|≤a|β2x|g2|4α+↓|4|≤a|β3x|4α+↓|4|≤a|β4 
 6986  |πof degree |→|¬E3.|'{A3}|≡2|≡7|≡.|9|4Otherwise 
 6990  there would be a nonzero polynomial |εf(q|βn,|4.|4.|4.|4,|4q
 6996  |β1,|4q|β0) |πwith integer coe∃cients such that 
 7002  |εqn|4αo↓|4f(q|βn,|4.|4.|4.|4,|4q|β1,|4q|β0)|4α=↓|40 
 7003  |πfor all sets (|εq|βn,|4.|4.|4.|4,|4q|β0) |πof 
 7008  real numbers. This cannot happen, since it is 
 7016  easy to prove by induction on |εn |πthat a nonzero 
 7026  polynomial always takes on some nonzero value. 
 7033  (However, this result is false for|ε⊂nite |π_elds 
 7040  in place of the real numbers.)|'{A3}|≡2|≡8|≡.|9|4The 
 7047  indeterminate quantities |ε|≤a|β1,|4.|4.|4.|4,|4|≤a|βs 
 7050  |πform an algebraic basis for |εQ[|≤a|β1,|4.|4.|4.|4,|4|≤a|β
 7055  s], |πwhere |εQ |πis the _eld of rational numbers. 
 7064  Since |εs|4α+↓|41 |πis greater than the number 
 7071  of elements in a basis, the polynomials |εf|βj(|≤a|β1,|4.|4.
 7078  |4.|4,|4|≤a|βs) |πare algebraically dependent; 
 7082  this means that that there is a nonzero polynomial 
 7091  |εg |πwith rational coe∃cients such that |εg{H11}({H9}f|β0(|
 7097  ≤a|β1,|4.|4.|4.|4,|4|≤a|βs),|4.|4.|4.|4,|4f|βs(|≤a|β1,|4.|4.
 7097  |4.|4,|4|≤a|βs){H11}){H9} |πis identically zero.|'
 7101  {A3}|≡2|≡9|≡.|9|4Given |εj|β0,|4.|4.|4.|4,|4j|βt|4|¬A|4|¬T0,
 7102  |41,|4.|4.|4.|4,|4n|¬Y, |πthere are nonzero polynomials 
 7107  with integer coe∃cients such that |εg|βj(q|βj|m0,|4.|4.|4.|4
 7112  ,|4q|βj|mt)|4α=↓|40 |πfor all (|εq|βn,|4.|4.|4.|4,|4q|β0) 
 7116  |πin |εR|βj, 1|4|¬E|4j|4|¬E|4m. |πThe product 
 7121  |εg|β1g|β2|4.|4.|4.|4g|βm |πis therefore zero 
 7125  for all (|εq|βn,|4.|4.|4.|4,|4q|β0) |πin |εR|β1|4|↔q|4αo↓|4α
 7129  o↓|4αo↓|4|↔q|4R|βm.|'{A3}|≡3|≡0|≡.|9|4|πStarting 
 7131  with the construction in Theorem M, we will prove 
 7140  that |εm|β2|4α+↓|4(1|4α_↓|4|≤d|β0|βm|m1) |πof 
 7143  the |ε|≤b|π's may e=ectively be eliminated: If 
 7150  |ε|≤u|βi |πcorresponds to a parameter multiplication, 
 7156  we have |ε|≤m|βi|4α=↓|4|≤b|β2|βi|βα_↓|β1|4α⊗↓|4(T|β2|βi|4α+↓
 7158  |4|≤b|β2|βi); |πadd |εc|≤b|β2|βi|βα_↓|β1|≤b|β2|βi 
 7161  |πto each |ε|≤b|βj |πfor which |εc|≤m|βi |πoccurs 
 7168  in |εT|βj, |πand replace |ε|≤b|β2|βi |πby zero. 
 7175  This removes one aprameter multiplication. If 
 7181  |ε|≤m|βi |πis the _rst chain multiplication, 
 7187  then |ε|≤m|βi |πis the _rst chain multiplication, 
 7194  then |ε|≤m|βi|4α=↓|4(|≤g|β1x|4α+↓|4|≤u|β1|4α+↓|4|≤b|β2|βi|βα
 7195  _↓|β1)|4α⊗↓|4(|≤g|β2|βx|4α+↓|4|≤u|β2|4α+↓|4|≤b|β2|βi), 
 7196  |πwhere |ε|≤g|β1, |≤g|β2, |≤u|β1, |≤u|β2 |πare 
 7202  polynomials in |ε|≤b|β1,|4.|4.|4.|4,|4|≤b|β2|βi|βα_↓|β2 
 7205  |πwith integer coe∃cients. Here |ε|≤u|β1 |πand 
 7211  |ε|≤u|β2 |π can be ``absorbed'' into |ε|≤b|β2|βi|βα_↓|β1 
 7218  |πand |ε|≤b|β2|βi, |πrespectively, so we may 
 7224  assume that |ε|≤u|β1|4α=↓|4|≤u|β2|4α=↓|40. |πNow 
 7228  add |εc|≤b|β2|βi|βα_↓|β1|≤b|β2|βi |πto each |ε|≤b|βj 
 7233  |πfor which |εc|≤m|βi |πoccurs in |εT|βj; |πadd 
 7240  |ε|≤b|β2|βi|βα_↓|β1|≤g|β2/|≤g|β1 |πto |ε|≤b|β2|βi; 
 7243  |πand set |ε|≤b|β2|βi|βα_↓|β1 |πto zero. The 
 7249  result set is unchanged by this elimination of 
 7257  |ε|≤b|β2|βi|βα_↓|β1, |πexcept for the values 
 7262  of |ε|≤a|β1,|4.|4.|4.|4,|4|≤a|βs |πsuch that 
 7266  |ε|≤g|β1 |πis zero. {H11}({H9}This proof is essentially 
 7273  due to V. Ya. Pan, |εRussian Mathematical Surveys 
 7281  |π|≡2|≡1 (1966), 105<136.{H11}){H9} The latter 
 7286  case can be handled as in the proof of Theorem 
 7296  A, since the polynomials with |ε|≤g|β1|4α=↓|40 
 7302  |πcan be evaluated by eliminating |ε|≤b|β2|βi 
 7308  (|πas in the _rst construction, where |ε|≤m|βi 
 7315  |πcorresponds to a parameter multiplication).|'
 7320  {A3}|≡3|≡1|≡.|9|4Otherwise we could add one parameter 
 7326  multiplication as a _nal step, and violate Theorem 
 7334  C. (The exercise is an improvement over Theorem 
 7342  A, in this special case since there are only 
 7351  |εn |πdegrees of freedom in the coe∃cients of 
 7359  a monic polynomial of degree |εn.)|'{A3}|≡3|≡2|≡.|9|4|≤l|β1|
 7365  4α=↓|4|≤l|β0|4α⊗↓|4|≤l|β0, |≤l|β2|4α=↓|4|≤a|β1|4α⊗↓|4|≤l|β1,
 7366   |≤l|β3|4α=↓|4|≤a|β2|4α+↓|4|≤l|β2, |≤l|β4|4α=↓|4|≤l|β3|4α⊗↓|
 7368  4|≤l|β1,|4|≤l|β5|4α=↓|4|≤a|β3|4α+↓|4|≤l|β4. |πWe 
 7370  need at least three multiplications to compute 
 7377  |εu|β4x|g4 (|πsee Section 4.6.3), and at least 
 7384  two additions by Theorem A.|'{A3}|≡3|≡3|≡.|9|4We 
 7390  must have |εn|4α+↓|41|4|¬E|42m|β1|4α+↓|4m|β2|4α+↓|4|≤d|β0|βm
 7392  |m1, |πand |εm|β1|4α+↓|4m|β2|4α=↓|4(n|4α+↓|41)/2; 
 7395  |πso there are no parameter multiplications. 
 7401  Now the _rst |ε|≤l|βi |πwhose leading coe∃cient 
 7408  (as a polynomial in |εx) |πis not an integer 
 7417  must be obtained by a chain addition; and there 
 7426  must be at least |εn|4α+↓|41 |πparameters, so 
 7433  there are at least |εn|4α+↓|41 |πparameter additions.|'
 7440  {A3}|≡3|≡4|≡.|9|4Transform the given chain step 
 7445  by step, and also de_ne the ``content'' |εc|βi 
 7453  |πof |ε|≤l|βi, |πas follows: (Intuitively, |εc|βi 
 7459  |πis the leading coe∃cient of |ε|≤l|βi.) |πDe_ne 
 7466  |εc|β0|4α=↓|41. |π(a) If the step has the form 
 7474  |ε|≤l|βi|4α=↓|4|≤a|βj|4α+↓|4|≤l|βk, |πreplace 
 7476  it by |ε|≤l|βi|4α=↓|4|≤b|βj|4α=↓|4|≤a|βj/c|βk; 
 7479  |πand de_ne |εc|βi|4α=↓|4c|βk. |π(b) |πIf the 
 7485  step has the form |ε|≤l|βi|4α=↓|4|≤a|βj|4α_↓|4|≤l|βk, 
 7490  |πreplace it by |ε|≤l|βi|4α=↓|4|≤b|βj|4α+↓|4|≤l|βk, 
 7494  |πwhere |ε|≤b|βj|4α=↓|4|≤a|βj/c|βk; |πand de_ne|εc|βi|4α=↓|4
 7497  c|βk. (|πb) If the step has the form |ε|≤l|βi|4α=↓|4|≤a|βj|4
 7505  α_↓|4|≤l|βk, |πreplace it by |ε|≤l|βi|4α=↓|4|≤b|βj|4α+↓|4|≤l
 7509  |βk, |πwhere |ε|≤b|βj|4α=↓|4|→α_↓|≤a|βj/c|βk; 
 7512  |πand de_ne |εc|βi|4α=↓|4|→α_↓c|βk. (|πc) If 
 7517  the step has the form |ε|≤l|βi|4α=↓|4|≤a|βj|4α⊗↓|4|≤l|βk, 
 7523  |πreplace it by |ε|≤l|βi|4α=↓|4|≤l|βk (|πthe 
 7528  step will be deleted later); and de_ne |εc|βi|4α=↓|4|≤a|βjc|
 7535  βk. |π(d) If the step has the form |ε|≤l|βi|4α=↓|4|≤l|βj|4α⊗
 7543  ↓|4|≤l|βk, |πleave it unchanged; and de_ne |εc|βi|4α=↓|4c|βj
 7549  c|βk.|'!!|1|1|πAfter this process is _nished, 
 7555  delete all steps of the form |ε|≤l|βi|4α=↓|4|≤l|βk, 
 7562  |πreplacing |ε|≤l|βi |πby |ε|≤l|βk |πin each 
 7568  future step which uses |ε|≤l|βj. |πThen add a 
 7576  _nal step |ε|≤l|βr|βα+↓|β1|4α=↓|4|≤b|4α⊗↓|4|≤l|βr, 
 7579  |πwhere |ε|≤b|4α=↓|4c|βr. |πThis is the desired 
 7585  scheme, since it is easy to verify that the new 
 7595  |ε|≤l|βi |πare just the old ones divided by the 
 7604  factor |εc|βi. |πThe |ε|≤b|π's are given functions 
 7611  of the |ε|≤a|π's; division by zero is no problem, 
 7620  because if any |εc|βk|4α=↓|40 |πwe must have 
 7627  |εc|βr|4α=↓|40 (|πhence the coe∃cient of |εx|gn 
 7633  |πis zero), or else |ε|≤l|βk |πnever contributes 
 7640  to the _nal result.|'{A3}|≡3|≡5|≡.|9|4Since there 
 7646  are at least _ve parameter steps, the result 
 7654  is trivial unless there is at least one parameter 
 7663  multiplication; considering the ways in which 
 7669  three multiplications can form |εu|β4x|g4, |πwe 
 7675  see that there must be one parameter multiplication 
 7683  and two chain multiplications. Therefore the 
 7689  four addition-subtractions must each be parameter 
 7695  steps, and exercise 34 applies. We can now assume 
 7704  that only additions are used, and that we have 
 7713  a chain to compute a general |εmonic |πfourth 
 7721  degree polynomial with |εtwo |πchain multiplications 
 7727  and four parameter additions. The only possible 
 7734  scheme of this type which calculates a fourth 
 7742  degree polynomial has the form|'{A9}|ε|≤l|β1|4α=↓|4|≤a|β1|4α
 7747  +↓|4|≤l|β0|;{A4}|≤l|β2|4α=↓|4|≤a|β2|4α+↓|4|≤l|β0|;
 7749  {A4}|≤l|β3|4α=↓|4|≤l|β1|4α⊗↓|4|≤l|β2|;{A4}|≤l|β4|4α=↓|4|≤a|β
 7750  3|4α+↓|4|≤l|β3|;{A4}|≤l|β5|4α=↓|4|≤a|β4|4α+↓|4|≤l|β3|;
 7752  {A4}|≤l|β6|4α=↓|4|≤l|β4|4α⊗↓|4|≤l|β5|;{A4}|≤l|β7|4α=↓|4|≤a|β
 7753  5|4α+↓|4|≤l|β6|;{A9}|πActually this chain has 
 7758  one addition too many, but any correct scheme 
 7766  can be put into this form if we restrict some 
 7776  of the |ε|≤a|π's to be the functions of the others. 
 7786  Now |ε|≤l|β7 |πhas the form (|εx|g2|4α+↓|4Ax|4α+↓|4B)(x|g2|4
 7791  α+↓|4Ax|4α+↓|4C)|4α+↓|4D|4α=↓|4x|g4|4α+↓|42Ax|g3|4α+↓|4(E|4α
 7791  +↓|4A|g2)x|g2|4α+↓|4EA|βx|4α+↓|4F, |πwhere |εA|4α=↓|4|≤a|β1|
 7793  4α+↓|4|≤a|β2, B|4α=↓|4|≤a|β1|4α+↓|4|≤a|β2, B|4α=↓|4|≤a|β1|≤a
 7795  |β2|4α+↓|4|≤a|β3, C|4α=↓|4|≤a|β1|≤a|β2|4α+↓|4|≤a|β4, 
 7797  D|4α=↓|4|≤a|β6, E|4α=↓|4B|4α+↓|4C, F|4α=↓|4BC|4α+↓|4D; 
 7800  |πand since this involves only three independent 
 7807  parameters it cannot represent a general monic 
 7814  fourth-degree polynomial.|'{A3}|ε|∂|≤l|β1|9|2|∂α=↓|4|≤a|β1|4
 7816  α+↓|4|≤l|β0!!!!|∂|≤l|β1|9|2|∂α=↓|4|≤a|β1|4α+↓|4|≤l|β0|;
 7817  {A4}|L|≤l|β2|Lα=↓|4|≤a|β2|4α+↓|4|≤l|β0|L|≤l|β2|Lα=↓|4|≤a|β2|
 7817  4α+↓|4|≤l|β0>{A4}|L|≤l|β3|Lα=↓|4|↔l|β1|4α⊗↓|4|≤l|β2|L|≤l|β3|
 7818  Lα=↓|4|≤l|β1|4α⊗↓|4|≤l|β2>{A4}|L|≤l|β4|Lα=↓|4|≤a|β3|4α+↓|4|≤
 7819  l|β0|L|≤l|β4|Lα=↓|4|≤a|β3|4α+↓|4|≤l|β3>{A4}|L|≤l|β5|Lα=↓|4|≤
 7820  a|β4|4α+↓|4|≤l|β3|L|≤l|β5|Lα=↓|4|≤a|β4|4α+↓|4|≤l|β3>
 7821  {A4}|L|≤l|β6|Lα=↓|4|≤l|β4|4α⊗↓|4|≤l|β5|L|≤l|β6|Lα=↓|4|≤l|β4|
 7821  4α⊗↓|4|≤l|β5>{A4}|L|≤l|β7|Lα=↓|4|≤a|β5|4α+↓|4|≤l|β6|L|≤l|β7|
 7822  Lα=↓|4|≤a|β5|4α+↓|4|≤l|β3>{A4}|L|≤l|β8|Lα=↓|4|≤a|β6|4α+↓|4|≤
 7823  l|β6|L|≤l|β8|Lα=↓|4|≤a|β6|4α+↓|4|≤l|β6>{A4}|L|≤l|β9|Lα=↓|4|≤
 7824  l|β7|4α⊗↓|4|≤l|β8|L|≤l|β9|Lα=↓|4|≤l|β7|4α⊗↓|4|≤l|β8>
 7825  {A4}|L|≤l|β1|β0|Lα=↓|4|≤a|β7|4α+↓|4|≤l|β9|L|≤l|β1|β0|Lα=↓|4|
 7825  ≤a|β7|4α+↓|4|≤l|β9>{A9}|πwhere, as in exercise 
 7830  35, an extra addition has been inserted to cover 
 7839  a more general case. Neither of these schemes 
 7847  can calculate a general sixth-degree monic polynomial, 
 7854  since the _rst case is a polynomial of the form|'
 7864  {A9}(|εx|g3|4α+↓|4Ax|g2|4α+↓|4Bx|4α+↓|4C)(x|g3|4α+↓|4Ax|g2|4
 7864  α+↓|4Bx|4α+↓|4D)|4α+↓|4E,|;{A9}|πand in the second 
 7869  case (cf. exercise 35) is a polynomial of the 
 7878  form|'{A9}{H11}({H9}|εx|g4|4α+↓|42Ax|g3|4α+↓|4(E|4α+↓|4A|g2)
 7879  x|g2|4α+↓|4EAx|4α+↓|4F{H11}){H9}(x|g2|4α+↓|4Ax|4α+↓|4G)|4α+↓
 7879  |4H;|;{A9}|πboth of these involve only _ve independent 
 7887  parameters.|'{A3}|≡3|≡7|≡.|9|4Let |εu|g[|g0|g](x)|4α=↓|4u|βn
 7889  x|gn|4α+↓|4u|βn|βα_↓|β1x|gn|gα_↓|g1|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓
 7889  |4u|β0, v|g[|g0|g](x)|4α=↓|4x|gn|4α+↓|4v|βn|βα_↓|β1x|gn|gα_↓
 7890  |g1|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|4v|β0. |πFor |ε1|4|¬E|4j|4|¬E|4
 7892  n, |πdivide |εu[|gj|gα_↓|g1|g](x) |πby the monic 
 7898  polynomial |εv|g[|gj|gα_↓|g1|g](x), |πobtaining 
 7901  |εu|g[|gj|gα_↓|g1|g](x)|4α=↓|4|≤a|βjv|g[|gj|gα_↓|g1|g](x)|4α
 7901  +↓|4|≤b|βjv|g[|gj|g](x). |πAssume that a monic 
 7906  polynomial |εv|g[|gj|g](x) |πof degree |εn|4α_↓|4j 
 7911  |πexists satisfying this relation; this will 
 7917  be true for almost all rational functions. Let 
 7925  |εu|g[|gj|g](x)|4α=↓|4v|g[|gj|gα_↓|g1|g](x)|4α_↓|4xv|g[|gj|g
 7925  ](x). |πThese de_nitions imply that deg(|ε|gu|g])|4|¬W|41, 
 7931  |πso we may let |ε|≤a|βn|βα+↓|β1|4α=↓|4u|g[|gn|g](x).|'
 7936  !!|1|1|πFor the given rational function we have|'
 7943  {A9}|ε|∂!|≤a|βj!|∂!|≤b|βj!|∂v|g[|gj|g](x)!|∂!|9u|g[|gj|g](x)
 7943  |9!|∂|;{A3}|>1|;2|;x|4α+↓|45|;3x|4α+↓|419|;>|>
 7951  3|;4|;1|;5|;>{A9}|πso |εu|g[|g0|g](x)/v|g[|g0|g](x)|4α=↓|41|
 7957  4α+↓|42/{H11}({H9}x|4α+↓|43|4α+↓|44/(x|4α+↓|45){H11}){H9}.|'
 7958  |ε|*/Notes:|\ |πA general rational function of 
 7964  the stated form has |ε2n|4α+↓|41 |π``degrees 
 7970  of freedom,'' in the sense that it can be shown 
 7980  to have |ε2n|4α+↓|41 |πessentially independent 
 7985  parameters. If we generalize polynomial chains 
 7991  to ``arithmetic chains,'' which allow division 
 7997  operations as well as addition, subtraction, 
 8003  and multiplication, we can obtain the following 
 8010  results with slight modi_cations to the proofs 
 8017  of Theorems A |πand M: |εAn arithmetic chain 
 8025  with q addition-subtraction steps has at most 
 8032  q|4α+↓|41 degrees of freedom. An arithmetic chain 
 8039  with m multiplication-division steps has at most 
 8046  2m|4α+↓|41 degrees of freedom. |πTherefore an 
 8052  arithmetic chain which computes almost all ration 
 8059  functions of the stated form must have at least 
 8068  |ε2n |πaddition-subtractions, and |εn |πmultiplication-divis
 8072  ions; the method in this exercise is ``optimal.''|'
 8080  |H=*?*?{U0}{H9L11M29}|πW58320#Computer|4Programming!(Knuth/Add
folio 835 galley 9
 8080  ison-Wesley)!Answers!f.|4835!g.|49d|'{A7}|≡3|≡8|≡.|9|4The 
 8082  theorem is certainly ture if |εn|4α=↓|40. |πAssume 
 8089  that |εn |πis positive, and that a polynomial 
 8097  chain computing |εP(x;|4u|β0,|4.|4.|4.|4,|4u|βn) 
 8100  |πis given, where each of the parameters |ε|≤a|βj 
 8108  |πhas been replaced by a real number. Let |ε|≤l|βi|4α=↓|4|≤l
 8116  |βj|4α⊗↓|4|≤l|βk |πbe the _rst chain multiplication 
 8122  step which involves one of |εu|β0,|4.|4.|4.|4,|4u|βn; 
 8128  |πsuch a step must exist because of the rank 
 8137  of |εA. |πWithout loss of generality, we may 
 8145  assume that |ε|≤l|βj |πinvolves |εu|βn; |πthus, 
 8151  |ε|≤l|βj |πhas the form |εh|β0u|β0|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|
 8155  4h|βnu|βn|4α+↓|4f(x), |πwhere |εh|β0,|4.|4.|4.|4,|4h|βn 
 8158  |πare real, |εh|βn|4|=|↔6α=↓|40, |πand |εf(x) 
 8163  |πis a polynomial with real coe∃cients. {H11}({H9}The 
 8170  |εh|π's and the coe∃cients of |εf(x) |πare derived 
 8178  from the values assigned to the |ε|≤a|π's.{H11}){H9}|'
 8185  !!|1|1Now change step |εi |πto |ε|≤l|βi|4α=↓|4|≤a|4α⊗↓|4|≤l|
 8190  βk, |πwhere |ε|≤a |πis an arbitrary real number. 
 8198  (We could take |ε|≤a|4α=↓|40; |πgeneral |ε|≤a 
 8204  |πis used here merely to show that there is a 
 8214  certain amount of ⊗exibility available in the 
 8221  proof.) Add further steps to calculate|'{A9}|ε|≤l|4α=↓|4{H11
 8227  }({H9}|≤a|4α_↓|4f(x)|4α_↓|4h|β0u|β0|4α_↓|4αo↓|4αo↓|4αo↓|4α_↓
 8227  |4h|βn|βα_↓|β1u|βn|βα_↓|β1{H11}){H9}/h|βn;|;{A9}|πthese 
 8229  new steps involve only additions and parameter 
 8236  multiplications (by suitable new parameters). 
 8241  Finally, replace |ε|≤l|βα_↓|βn|βα_↓|β1|4α=↓|4u|βn 
 8244  |πeverywhere in the chain by this new element 
 8252  |ε|≤l. |πThe result is a chain which calculates|'
 8260  {A9}|εQ(x;|4u|β0,|4.|4.|4.|4,|4u|βn|βα_↓|β1)|4α=↓|4P(x;|4u|β
 8260  0,|4.|4.|4.|4,|4u|βn|βα_↓|β1,|4(|≤a|4α_↓|4f(x)|4α_↓|4h|β0u|β
 8260  0|4α_↓|4αo↓|4αo↓|4αo↓|4α_↓|4h|βn|βα_↓|β1u|βn|βα_↓|β1)/h|βn{H
 8260  11}){H9};|'{A9}|πand this chain has one less 
 8267  chain multiplication. The proof will be complete 
 8274  if we can show that |εQ |πsatis_es the hypotheses. 
 8283  The quantity {H11}({H9}|ε|≤a|4α_↓|4f(x){H11}){H9}/h|βn 
 8286  |πleads to a possibly increased value of |εm, 
 8294  |πand a new vector |εB|¬S. |πIf the columns of 
 8303  |εA |πare |εA|β0,|4A|β1,|4.|4.|4.|4,|4A|βn |π(these 
 8307  vectors being linearly independent over the reals), 
 8314  the new matrix |εA|¬S |πcorresponding to |εQ 
 8321  |πhas the column vectors|'{A9}|εA|β0|4α_↓|4(h|β0/h|βn)A|βn,!
 8325  !.|4.|4.|4,!!A|βn|βα_↓|β1|4α_↓|4(h|βn|βα_↓|β1/h|βn)A|βn,|;
 8326  {A9}|πplus perhaps a few rows of zeros to account 
 8335  for an increased value of |εm, |πand these columns 
 8344  are clearly also linearly independent. By induction, 
 8351  the chain which computes |εQ |πhas at least |εn|4α_↓|41 
 8360  |πchain multiplications, so the original chain 
 8366  has at least |εn.|'!!|1|1|π(This proof of Ostrowski's 
 8374  conjecture is taken from Garsia's unpublished 
 8380  notes. An independent proof, which shows further 
 8387  that the possibility of division gives no improvement, 
 8395  was published by Pan in his survey paper cited 
 8404  in connection with exercise 30. Generalizations 
 8410  to the computation of several polynomials in 
 8417  several variables, with and without various kinds 
 8424  of preconditioning, have been given by S. Winograd, 
 8432  |εComm. Pure and Applied Math. |π|≡2|≡3 (1970), 
 8439  165<179. In particular, Winograd showed that 
 8445  at least |εmn |πmultiplications are needed to 
 8452  multiply a general |εm|4α⊗↓|4n |πmatrix by the 
 8459  vector (|εx|β1,|4.|4.|4.|4,|4x|βn), |πand at 
 8463  least |"l|f1|d32|)|εmn|"L |πof these do not depend 
 8470  only on the entries of the matrix or only on 
 8480  the entries of the vector.|'{A3}|≡3|≡9|≡.|9|4By 
 8486  induction on |εm. |πLet |εw|βm(x)|4α=↓|4x|g2|gm|4α+↓|4u|β2|β
 8490  m|βα_↓|β1x|g2|gm|gα_↓|g1|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|4u|β0, 
 8491  w|βm|βα_↓|β1(x)|4α=↓|4x|g2|gm|gα_↓|g2|4α+↓|4v|β2|βm|βα_↓|β3x
 8491  |g2|gm|gα_↓|g3|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|4v|β0, 
 8492  a|4α=↓|4|≤a|β1|4α+↓|4|≤g|βm, b|4α=↓|4|≤a|βm, 
 8494  |πand let |εf(r)|4α=↓|4|¬K|βi|β,|βj|β|¬R|β0|4(|→α_↓1)|gi|gα+
 8496  ↓|gj(|fiα+↓j|d5j|))u|βr|βα+↓|βi|βα+↓|β2|βja|gib|gj. 
 8497  |πIt follows that |εv|βr|4α=↓|4f(r|4α+↓|42) |πfor 
 8502  |εr|4|¬R|40, |πand |ε|≤d|βm|4α=↓|4f(1). |πIf 
 8506  |ε|≤d|βm|4α=↓|40 |πand |εa |πis given, we have 
 8513  a polynomial of degree |εm|4α_↓|41 |πin |εb, 
 8520  |πwith leading coe∃cient |→|¬N(|εu|β2|βm|βα_↓|β1|4α_↓|4ma)|4
 8523  α=↓|4|→|¬N(|≤g|β2|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|4|≤g|βm|4α_↓|4m|≤
 8523  g|βm).|'!!|1|1|πIn Motzkin's unpublished notes 
 8528  he arranged to make |ε|≤d|βk|4α=↓|40 |πalmost 
 8534  always, by choosing |ε|≤g|π's so that this leading 
 8542  coe∃cient is |=|↔6|→α=↓0 when |εm |πis even and 
 8550  |→α=↓0 when |εm |πis odd; then we almost always 
 8559  can let |εb |πbe a (real) root of an odd-degree 
 8569  polynomial.|'{A3}|≡4|≡0|≡.|9|4No; S. Winograd 
 8573  found a way to compute all polynomials of degree 
 8582  13 with only 7 (possibly complex) multiplications 
 8589  [|εComm. Pure and Applied Math. |π|≡2|≡5 (1972), 
 8596  455<457]. L. Revah found schemes which evaluate 
 8603  almost all polynomials of degree |εn|4|¬R|49 
 8609  |πwith |ε|"ln/2|"L|4α+↓|41 (|πpossibly complex) 
 8613  multiplications |ε[SIAM J. Computing |π|≡4 (1975), 
 8619  381<392]; for |εn|4α=↓|49 |πher scheme is |εu(x)|4α=↓|4{H11}
 8625  ({H9}v(x)(x|4α+↓|4|≤a)|4α_↓|4{H11}({H9}v(x)|4α+↓|4|≤b{H11}){
 8625  H9}(x|4α+↓|4|≤g)|4α+↓|4|≤d{H11}){H9}|4αo↓|4{H11}({H9}v(x)(x|
 8625  4α+↓|4|≤a)|4α+↓|4|≤e{H11}){H9}|4α+↓|4|≤j, |πwhere 
 8627  |εv(x) |πis a monic polynomial of degree 4. By 
 8636  appending su∃ciently many additions (cf. exercise 
 8642  39), the ``almost all'' and ``possibly complex'' 
 8649  provisos disappear.|'{A3}|≡4|≡1|≡.|9|4|εa(c|4α+↓|4d)|4α_↓|4(
 8651  a|4α+↓|4b)d|4α+↓|4i{H11}({H9}a(c|4α+↓|4d)|4α+↓|4(b|4α_↓|4a)c
 8651  {H11}){H9}; |πor (cf. Eq. 4.3.3 with |ε2|gn |πreplaced 
 8659  by |εi*3) ac|4α_↓|4bd|4α+↓|4i{H11}({H9}(a|4α+↓|4b)(c|4α+↓|4d)
 8661  |4α_↓|4ac|4α_↓|4bd{H11}){H9}; |πetc. [Beware 
 8664  numerical instability.]|'{A3}|π|≡4|≡2|≡.|9|4a)|9Let 
 8667  |ε|≤p|β1,|4.|4.|4.|4,|4|≤p|βm |πbe the |ε|≤l|βi|π's 
 8671  which correspond to chain multiplications; then 
 8677  |ε|≤p|βi|4α=↓|4P|β2|βi|βα_↓|β1|4α⊗↓|4P|β2|βi 
 8678  |πand |εu(x)|4α=↓|4P|β2|βm|βα+↓|β1, |πwhere each 
 8682  |εP|βj |πhas the form |ε|≤b|βj|4α+↓|4|≤b|βj|β0x|4α+↓|4|≤b|βj
 8686  |β1|≤p|β1|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|4|≤b|βj|βr|β(|βj|β)|≤p|βr
 8686  |β(|βj|β), |πwhere |εr(j)|4|¬E|4|"pj/2|"P|4α_↓|41 
 8689  |πand each of the |ε|≤b|βj |πand |ε|≤b|βj|βk 
 8696  |πis a polynomial in the |ε|≤a|π's with integer 
 8704  coe∃cients. We can systematically modify the 
 8710  chain (cf. exercise 30) so that |ε|≤b|βj|4α=↓|40 
 8717  |πand |ε|≤b|βj|βr|β(|βj|β)|4α=↓|41, |πfor 1|4|¬E|4|εj|4|¬E|4
 8720  2m; |πfurthermore we can assume that |ε|≤b|β3|β0|4α=↓|40. 
 8727  |πThe result set now has at most |εm|4α+↓|41|4α+↓|4|¬K|ur|)1
 8734  |¬Ej|¬E2m|)|4(|"pj/2|"P|4α_↓|41)|4α=↓|4m|g2|4α+↓|41 
 8735  |πdegrees of freedom.|'!!|1|1b)|9Any such polynomial 
 8741  chain with at most |εm |πchain multiplications 
 8748  can be simulated by one with the form considered 
 8757  in (a), except that now we let |εr(j)|4α=↓|4|"pj/2|"P|4α_↓|4
 8764  1 |πfor 1|4|¬E|4j|4|¬E|4|ε2m|4α+↓|41, |πand we 
 8769  do not assume that |ε|≤b|β3|β0|4α=↓|40 |πor that 
 8776  |ε|≤b|βj|βr|β(|βj|β)|4α=↓|41 |πfor |εj|4|¬R|43. 
 8779  |πThis single canonical form involves |εm|g2|4α+↓|42m 
 8785  |ε|≤b|π's. As the |ε|≤a|π's run thru all integers 
 8793  and as we run through all chains, the |ε|≤b|π's 
 8802  run through at most |ε2|gm|i2|gα+↓|g2|gm |πsets 
 8808  of values mod|42, hence the result set does also. 
 8817  In order to obtain all |ε2|gn |πpolynomials of 
 8825  degree |εn |πwith 0<1 coe∃cients, we need |εm|g2|4α+↓|42m|4|
 8832  ¬R|4n.|'!!|1|1|πc)|9Set |εm|4|¬L|4|"l|¬H|v4n|)|"L 
 8835  |πand compute |εx|g2,|4x|g3,|4.|4.|4.|4,|4x|gm. 
 8838  |πLet |εu(x)|4α=↓|4u|βm|βα+↓|β1(x)x|g(|gm|gα+↓|g1|g)|gm|4α+↓
 8839  |4αo↓|4αo↓|4αo↓|4α+↓|4u|β1(x)x|gm|4α+↓|4u|β0(x), 
 8840  |πwhere each |εu|βj(x) |πis a polynomial of degree 
 8848  |ε|→|¬Em |πwith integer coe∃cients (hence it 
 8854  can be evaluated without any more multiplications). 
 8861  Now evaluate |εu(x) |πby rule (2) as a polynomial 
 8870  in |εx|gm |πwith known coe∃cients. (The number 
 8877  of additions used is approximately the sum of 
 8885  the absolute values of the coe∃cients, so this 
 8893  algorithm is e∃cient on 0<1 polynomials. Paterson 
 8900  and Stockmeyer also gave another algorithm which 
 8907  uses about |ε|¬H|v42n|) |πmultiplications.|'!!|1|1!!|1|1Refe
 8911  rence: |εIEEE Symp. Switching and Automata Theory 
 8918  |π|≡1|≡2 (1971), 140<143. R. J. Lipton has found 
 8926  0<1 polynomials that require |εc|=|g4|¬H|v4n|)/|πlog|4|εn 
 8931  |πchain multiplications even after complex preconditioning 
 8937  [|εIEEE Symp. Found. Comp. Sci. |π|≡1|≡6 (1975), 
 8944  6<10].|'{A3}|≡4|≡3|≡.|9|4When |εa|βi|4α=↓|4a|βj|4α+↓|4a|βk 
 8947  |πis a step in some optimal addition chain for 
 8956  |εn|4α+↓|41, |πcompute |εx|gi|4α=↓|4x|gjx|gk 
 8959  |πand |εp|βi↓α=↓|4p|βkx|gj|4α+↓|4p|βj, |πwhere 
 8962  |εp|βi|4α=↓|4x|gi|gα_↓|g1|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|4x|4α+↓|4
 8962  1; |πomit the _nal calculation of |εx|gn|gα+↓|g1. 
 8969  |πWe save one multiplication whenever |εa|βk|4α=↓|41, 
 8975  |πin particular when |εi|4α=↓|41. |π({H9}|πCf. 
 8980  exercise 4.6.3<31 with |ε|≤e|4α=↓|4|f1|d32|).)|'
 8984  {A24}{H10}|∨S|∨E|∨C|∨T|∨I|∨O|∨N |∨4|∨.|∨7|'{A12}{H9}|9|1|≡1|
 8986  ≡.|9|4Find the _rst nonzero coe∃cient |εV|βm, 
 8992  |πas in (4), and divide both |εU(z) |πand |εV(z) 
 9001  |πby shifting |εz|gm (|πshifting the coe∃cients 
 9007  |εm |πplaces to the left). The quotient will 
 9015  be a power series i= |εU|β0|4α=↓|4αo↓|4αo↓|4αo↓|4α=↓|4U|βm|β
 9020  α_↓|β1|4α=↓|40.|'{A3}|9|1|≡2|≡.|9|4|εV|urnα+↓1|)0|)W|βn|4α=↓
 9021  |4V|urn|)0|)U|βn|4α_↓|4(V|ur1|)0|)W|β0)(V|urnα_↓1|)0|)V|βn)|
 9021  4α_↓|4(V|ur2|)0|)W|β1)(V|urnα_↓2|)0|)V|βn|βα_↓|β1)|4α_↓|4αo↓
 9021  |4αo↓|4αo↓|4α_↓|4(V|urn|β0|βW|βn|βα_↓|β1)(V|ur0|)0|)V|β1). 
 9022  |πThus, we start by replacing |ε(U|βj,|4V|βj) 
 9028  |πby (|εV|urj|)0|)U|βj,|4V|urjα_↓1|)0|)V|βj) 
 9030  |πfor |εj|4|¬R|41, |πthen set |εW|βn|4|¬L|4U|βn|4α_↓|4|¬K|ur
 9034  |)0|¬Ek|¬Wn|)|4W|βkV|βn|βα_↓|βk |πfor |εn|4|¬R|40, 
 9037  |π_nally replace |εW|βj |πby |εW|βj/V|urjα+↓1|)0|) 
 9042  |πfor |εj|4|¬R|40. |πSimilar techniques can be 
 9048  used for other algorithms in this section.|'{A3}|9|1|≡3|≡.|9
 9055  |4Yes. When |ε|≤a|4α=↓|40, |πit is easy to prove 
 9063  by induction that |εW|β1|4α=↓|4W|β2|4α=↓|4αo↓|4αo↓|4αo↓|4α=↓
 9066  |40. |πWhen |ε|≤a|4α=↓|41, |πwe _nd |εW|βn|4α=↓|4V|βn, 
 9072  |πby the ``cute'' identity|'{A9}|↔k|uc|)|ε1|¬Ek|¬En|)|1|1|↔a
 9076  |(k|4α_↓|4(n|4α_↓|4k)|d2n|)|↔s|1V|βkV|βn|βα_↓|βk|4α=↓|4V|β0V
 9076  |βn.|;{A9}|9|1|≡4|≡.|9|4|πIf |εW(z)|4α=↓|4e|gV|g(|gz|g), 
 9079  |πthen |εW|¬S(z)|4α=↓|4V|¬S(z)W(z); |πwe _nd 
 9083  |εW|β0|4α=↓|41, |πand|'{A9}|εW|βn|4α=↓|4|↔k|uc|)1|¬Ek|¬En|)|
 9085  4|(k|d2n|)|4V|βkW|βn|βα_↓|βk,!!|πfor!!|εn|4|¬R|41.|;
 9086  {A9}|πIf |εW(z)|4α=↓|4|πln{H11}({H9}|ε1|4α+↓|4V(z){H11}){H9}
 9087  , |πthen |εW|¬S(z)|4α+↓|4W|¬S(z)V(z)|4α=↓|4V|¬S(z); 
 9090  |πthe rule is |εW|β0|4α=↓|40, |πand |εW|β1|4α+↓|42W|β2z|4α+↓
 9095  |4αo↓|4αo↓|4αo↓|4α=↓|4V|¬S(z)/{H11}({H9}1|4α+↓|4V(z){H11}){H
 9095  9}.|'!!|1|1(|πBy exercise 6, the logarithm can 
 9102  be obtained to order |εn |πin |εO(n|4|πlog|4|εn) 
 9109  |πoperations. R. P. Brent observes that exp{H11}({H9}|εV(z){
 9115  H11}){H9} |πcan also be calculated with this 
 9122  asymptotic speed by applying Newton's method 
 9128  to |εf(x)|4α=↓|4|πln|4|εx|4α_↓|4V(z); |πtherefore 
 9131  {H11}({H9}1|4α+↓|4|εV(z){H11}){H9}|g|≤a|4α=↓|4|πexp{H11}({H9
 9131  }|ε|≤a|4|πln(1|4α+↓|4|εV(z)){H11}){H9} |πis |εO(n|4|πlog|4|ε
 9133  n) |πtoo. |ε[Analytic Computational Complexity, 
 9138  |πed. by J. F. Traub (New York: Academic Press, 
 9147  1975), !|4|4|4<!|4|4|4.])|'{A3}|9|1|≡5|≡.|9|4We 
 9150  get the original series back again. This can 
 9158  be used to test a reversion algorithm.|'{A3}|9|1|≡6|≡.|9|4|ε
 9165  |≤'(x)|4α=↓|4x|4α+↓|4x{H11}({H9}1|4α_↓|4xV(z){H11}){H9}, 
 9166  |πcf. Algorithm 4.3.3R. Thus after |εW|β0,|4.|4.|4.|4,|4W|βN
 9171  |βα_↓|β1 |πare known, we input |εV|βN,|4.|4.|4.|4,|4V|β2|βN|
 9176  βα_↓|β1, |πcompute (|εW|β0|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|4W|βN|βα
 9178  _↓|β1z|gN|gα_↓|g1)(V|β0|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|4V|β2|βN|βα
 9178  _↓|β1z|g2|gN|gα_↓|g1)|4α=↓|41|4α+↓|4R|β0z|gN|4α+↓|4αo↓|4αo↓|
 9178  4αo↓|4α+↓|4R|βN|βα_↓|β1z|g2|gN|gα_↓|g1|4α+↓|4O(z|g2|gN), 
 9179  |πand determine |εW|βN,|4.|4.|4.|4,|4W|β2|βN|βα_↓|β1 
 9182  |πby the formula |εW|βN|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|4W|β2|βN|βα
 9185  _↓|β1z|gN|gα_↓|g1|4α=↓|4|→α_↓(W|β0|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|
 9185  4W|βN|βα_↓|β1z|gN|gα_↓|g1)(R|β0|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|4R|
 9185  βN|βα_↓|β1z|gN|gα_↓|g1)|4α+↓|4O(z|gN). [Numer. 
 9187  Math. |≡2|≡2 (1974), 341<348; |πthis algorithm 
 9193  was, in essence, _rst published by M. Sieveking, 
 9201  |εComputing |≡1|≡0 (|π1972), 153<156.] Note that 
 9207  the total time for |εN |πcoe∃cients is |εO(N|4|πlog|4|εN) 
 9215  |πif we use ``fast'' polynomial multiplication 
 9221  (exercise 4.6.4<14).|'{A3}|9|1|≡7|≡.|9|4|εW|βn|4α=↓|4(|fmk|d
 9223  5k|))/n |πwhen |εn|4α=↓|4(m|4α_↓|41)k|4α+↓|41, 
 9226  |πotherwise 0. (Cf. exercise 2.3.4.4<11.)|'|Ha*?{U0}{H9L11M29
folio 842 galley 10
 9231  }|πW58320#Computer|4Programming!(Knuth/Addison-Wesley)!Answe
 9231  rs!f.|4842!g.|410d|'{A6}|9|1|≡8|≡.|9|4Input |εG|β1 
 9234  |πin step L1, and |εG|βn |πin step L2. In step 
 9244  L4, the output should be (|εU|βn|βα_↓|β1G|β1|4α+↓|42U|βn|βα_
 9249  ↓|β2G|β2|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|4nU|β0G|βn)/n. 
 9250  (|πThe running time of the order |εN|g3 |πalgorithm 
 9258  is hereby increased by only order |εN|g2. |πThe 
 9266  value |εW|β1|4α=↓|4G|β1 |πmight be output in 
 9272  step L1.)|'!!|1|1|ε|*/Note:|\ |πAlgorithms T and 
 9278  N determine |εV|gα_↓|g1{H11}({H9}U(z){H11}); 
 9281  |πthe algorithm in this exercise determines |εG{H11}({H9}V|g
 9287  α_↓|g1(z){H11}){H9}, |πwhich is somewhat di=erent. 
 9292  Of course, the results can all be obtained by 
 9301  a sequence of operations of reversion and composition 
 9309  (exercise 11), but it is helpful to have more 
 9318  direct algorithms for each case.|'{A3}|9|1|≡9|≡.|E|'
 9324  |\∂!!|2|hT|β1|βn|n|∂!n|4α=↓|41!|∂!n|4α=↓|42!|∂!n|4α=↓|43!|∂!
 9324  n|4α=↓|44!|∂!n|4α=↓|45!|∂|'{A3}|>!!|2T|β1|βn|'
 9327  1|;1|;2|;5|;14|;>|>!!|2T|β2|βn|'|'1|;2|;5|;14|;
 9340  >|>!!|2T|β3|βn|'|'|'1|;3|;|9|19|;>|>!!|2T|β4|βn|'
 9351  |'|'|'1|;|9|14|;>|>!!|2T|β5|βn|'|'|'|'|'|9|11|;
 9364  >{A3}|π|≡1|≡0|≡.|9|4Form |εy|g1|g/|g|≤a|4α=↓|4x(1|4α+↓|4a|β1
 9366  x|4α+↓|4a|β2x|g2|4α+↓|4αo↓|4αo↓|4αo↓)|g1|g/|g|≤a|4α=↓|4x(1|4
 9366  α+↓|4c|β1x|4α+↓|4c|β2x|g2|4α+↓|4αo↓|4αo↓|4αo↓) 
 9367  |πby means of (9); |πthen revert the latter series. 
 9376  (See the remarks following Eq. 1.2.11.3<11.)|'
 9382  {A3}|≡1|≡1|≡.|9|4Set |εW|β0|4|¬L|4U|β0, |πand 
 9385  set |ε(T|βk,|4W|βk)|4|¬L|4(V|βk,|40) |πfor 1|4|¬E|4|εk|4|¬E|
 9388  4N. |πThen for |εn|4α=↓|41, 2,|4.|4.|4.|4,|4N, 
 9393  |πdo the following: Set |εW|βj|4|¬L|4W|βj|4α+↓|4U|βnT|βj 
 9398  |πfor |εn|4|¬E|4j|4|¬E|4N; |πand then set |εT|βj|4|¬L|4T|βj|
 9403  βα_↓|β1V|β1|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|4T|βnV|βj|βα_↓|βn 
 9404  |πfor |εj|4α=↓|4N, N|4α_↓|41,|4.|4.|4.|4,|4n|4α+↓|41.|'
 9407  !!|2|πHere |εT(z) |πrepresents |εV(z)|gn. |πAn 
 9412  |εon-line |πpower-series algorithm for this problem, 
 9418  |πanalogous to Algorithm T, could be constructed, 
 9425  but it would require about |εN|g2/2 |πstorage 
 9432  locations. There is also an on-line algorithm 
 9439  which solves this exercise and needs only |εO(N) 
 9447  |πstorage locations: |πWe may assume that |εV|β1|4α=↓|41, 
 9454  |πif |εU|βk |πis replaced by |εU|βkV|urk|)1|) 
 9460  |πand |εV|βk |πis replaced by |εV|βk/V|β1 |πfor 
 9467  all |εk. |πThen we may revert |εV(z) |πby Algorithm 
 9476  L, using it soutput as input to the algorithm 
 9485  of exercise 8 with |εG|β1|4α=↓|4U|β1, G|β2|4α=↓|4U|β2, 
 9491  |πetc., thus computing |εU{H11}({H9}(V|gα_↓|g1)|gα_↓|g1(z){H
 9494  11}){H9}|4α_↓|4U|β0.|'!!|2|πBrent and Kung have 
 9499  constructed several algorithms which are asymptotically 
 9505  faster. For example, we can evaluate |εU(x) |πfor 
 9513  |εx|4α=↓|4V(z) |πby a slight variant of exercise 
 9520  4.6.4<42(c), doing about 2|¬H|v4|εn|) |πchain 
 9525  multiplications of cost |εM(n) |πand about |εn 
 9532  |πparameter multiplications of cost |εn; |πthe 
 9538  total time is therefore |εO{H11}({H9}|¬H|v4n|)M(n)|4α+↓|4n|g
 9542  2{H11}){H9}|4α=↓|4O(n|g2). |πA still faster method 
 9547  can be based on the identity |εU{H11}({H9}V|β0(z)|4α+↓|4z|gm
 9553  V|β1(z){H11}){H9}|4α=↓|4U{H11}({H9}V|β0(z){H11}){H9}|4α+↓|4z
 9553  |gmU|¬S{H11}({H9}V|β0(z){H11}){H9}V|β1(z)|4α+↓|4z|g2|gmU|≤C{
 9553  H11}({H9}V|β0(z){H11}){H9}V|β1(z){H11}){H9}V|β1(z)|g2/2*3|4α+
 9553  ↓|4αo↓|4αo↓|4αo↓|4, |πextending to about |εn/m 
 9558  |πterms, where we choose |εm|4|¬V|4|¬H|v4n/|πlog|4|εn|); 
 9563  |πthe _rst term |εU{H11}({H9}V|β0(z){H11}){H9} 
 9567  |πis evaluated in |εO{H11}({H9}mn(|πlog|4|εn){H11}){H9}|g2) 
 9571  |πoperations using a method somewhat like that 
 9578  in exercise 4.6.4<43, |πand {H11}({H9}since we 
 9584  can go from |εU|g(|gk|g){H11}({H9}V|β0(z){H11}){H9} 
 9588  |πto |εU|g(|gk|gα+↓|g|≤n|g){H11}({H9}V|β0(z){H11}){H9} 
 9590  |πin |εO(n|4|πlog|4|εn) |πoperations by di=erentiating 
 9595  and dividing by |εV|ur|↔0|)1|)(z){H11}){H9} |πthe 
 9600  entire procedure takes |εO{H11}({H9}mn(|πlog|4|εn)|g2|4α+↓|4
 9603  (n/m)n|4|πlog|4|εn)|4α=↓|4O(n|4|πlog|4|εn)|g3|g/|g2 
 9604  |πoperations.|'{A3}|≡1|≡2|≡.|9|4Polynomial division 
 9607  is trivial unless |εm|4|¬R|4n|4|¬R|41. |πAssuming 
 9612  the latter, the equation |εu(x)|4α=↓|4q(x)v(x)|4α+↓|4r(x) 
 9617  |πis equivalent to |εU(z)|4α=↓|4Q(z)V(z)|4α+↓|4z|gm|gα_↓|gn|
 9620  gα+↓|g1R(z) |πwhere |εU(x)|4α=↓|4x|gmu(x|gα_↓|g1), 
 9623  V(x)|4α=↓|4x|gnv(x|gα_↓|g1), Q(x)|4α=↓|4x|gm|gα_↓|gnq(x|gα_↓
 9624  |g1), R(x)|4α=↓|4x|gn|gα_↓|g1r(x|gα_↓|g1) |πare 
 9627  the ``reverse'' polynomials of |εu, v, q, |πand 
 9635  |εr.|'!!|1|1|πTo _nd |εq(x) |πand |εr(x), |πcompute 
 9642  the _rst |εm|4α_↓|4n|4α+↓|41 |πcoe∃cients of 
 9647  the power series |εU(z)/V(z)|4α=↓|4W(z)|4α+↓|4O(z|gm|gα_↓|gn
 9650  |gα+↓|g1); |πthen compute the power series |εU(z)|4α_↓|4V(z)
 9656  W(z), |πwhich has the form |εz|gm|gα_↓|gn|gα+↓|g1T(z) 
 9662  |πwhere |εT(z)|4α=↓|4T|β0|4α+↓|4T|β1z|4α+↓|4αo↓|4αo↓|4αo↓|4.
 9663   |πNote that |εT|βj|4α=↓|40 |πfor all |εj|4|¬R|4n; 
 9670  |πhence |εQ(z)|4α=↓|4W(z) |πand |εR(z)|4α=↓|4T(z) 
 9674  |πsatisfy the requirements.|'{A25}{H11}|!A|1|1|!P|1|1|!P|1|1
 9677  |!E|1|1|!N|1|1|!D|1|1|!I|1|1|!X|4|4|4|!A|'{A49}{A9}{H20L22}|
 9678  ∨T|∨A|∨B|∨L|∨E|∨S|9|∨O|∨F|?|∨N|∨U|∨M|∨E|∨R|∨I|∨C|∨A|∨L|9|∨Q|
 9679  ∨U|∨A|∨N|∨T|∨I|∨T|∨I|∨E|∨S|?{A12}{H8L10}|∨T|∨a|∨b|∨l|∨e|4|4|
 9680  ∨1|;{A6}{H9L11}{M22}QUANTITIES WHICH ARE FREQUENTLY 
 9685  USED IN STANDARD SUBROUTINES AND IN ANALYSIS 
 9692  OF COMPUTER PROGRAMS. (40 DECIMAL PLACES)|;|J#>
 9699  {M32}|ε|h|¬H|≤p|3α=↓|3|≤)(1/2)|4|∂α=↓|4!!|E|n|'
 9700  | |¬H|v42|)|4|Lα=↓|41.41421|935623!73095|904880|916887|92420
 9700  9|969807|985697|→α_↓>| |¬H|v43|)|4|Lα=↓|41.73205|908075|9688
 9701  77|929352|974463|941505|987236|969428|→α+↓>| |¬H|v45|)|4|Lα=
 9702  ↓|42.23606|979774|999789|969640|991736|968731|917623|954406|
 9702  →α+↓>| |¬H|v410|)|4|Lα=↓|43.16227|976601|968379|933199|98893
 9703  5|944432|971853|937196|→α_↓>| |¬H|v42|)|4|Lα=↓|41.25992|9104
 9704  98|994873|916476|972106|907278|922835|905703|→α_↓>
 9705  | |¬H|v43|)|4|Lα=↓|41.44224|995703|907408|938232|916383|9107
 9705  80|910958|983919|→α_↓>| |¬H|v42|)|4|Lα=↓|41.18920|971150|902
 9706  721|906671|974999|970560|947591|952930|→α_↓>| |πln|42|4|Lα=↓
 9707  |40.69314|971805|959945|930941|972321|921458|917656|980755|→
 9707  α+↓>| ln|43|4|Lα=↓|41.09861|922886|968109|969139|952452|9369
 9708  22|952570|946475|→α_↓>| ln|410|4|Lα=↓|42.30258|950929|994045
 9709  |968401|979914|954684|936420|976011|→α+↓>| 1/ln|42|4|Lα=↓|41
 9710  .44269|950408|988963|940735|999246|981001|989213|974266|→α+↓
 9710  >| 1/ln|410|4|Lα=↓|40.43429|944819|903251|982765|911289|9189
 9711  16|960508|922944|→α_↓>| |ε|≤p|4|Lα=↓|43.14159|926535|989793|
 9712  923846|926433|983279|950288|941972|→α_↓>| 1|¬P|3α=↓|3|≤p/180
 9713  |4|Lα=↓|40.01745|932925|919943|929576|992369|907684|988612|9
 9713  71344|→α+↓>| 1/|≤p|4|Lα=↓|40.31830|998861|983790|967153|9776
 9714  75|926745|902872|940689|→α+↓>| |≤p|g2|4|Lα=↓|49.86960|944010
 9715  |989358|961883|944909|999876|915113|953137|→α_↓>
 9716  | |¬H|v4|≤p|)|3|4α=↓|3|≤)(1/2)|4|Lα=↓|41.77245|938509|905516
 9716  |902729|981674|983341|914518|927975|→α+↓>| |≤)(1/3)|4|Lα=↓|4
 9717  2.67893|985347|907747|963365|956929|940974|967764|941287|→α_
 9717  ↓>| |≤)(2/3)|4|Lα=↓|41.35411|979394|926400|941694|952880|928
 9718  154|951378|955193|→α+↓>| |εe|4|Lα=↓|42.71828|918284|959045|9
 9719  23536|902874|971352|966249|977572|→α+↓>| 1/e|4|Lα=↓|40.36787
 9720  |994411|971442|932159|955237|970161|946086|974458|→α+↓>
 9721  | e|g2|4|Lα=↓|47.38905|960989|930650|922723|904274|960575|90
 9721  0781|931803|→α+↓>| |≤g|4|Lα=↓|40.57721|956649|901532|986060|
 9722  965120|990082|940243|910422|→α_↓>| |πln|4|ε|≤p|4|Lα=↓|41.144
 9723  72|998858|949400|917414|934273|951353|905871|916473|→α_↓>
 9724  | |≤f|4|Lα=↓|41.61803|939887|949894|984820|945868|934365|963
 9724  811|977203|→α+↓>| e|g|≤g|4|Lα=↓|41.78107|924179|990197|99852
 9725  3|965041|903107|917954|991696|→α+↓>| e|g|≤p|g/|g4|4|Lα=↓|42.
 9726  19328|900507|938015|945655|997696|959278|973822|934616|→α+↓>
 9727  | |πsin|41|4|Lα=↓|40.84147|909848|907896|950665|925023|92163
 9727  0|929899|996226|→α_↓>| cos|41|4|Lα=↓|40.54030|923058|968139|
 9728  971740|909366|907442|997660|937323|→α+↓>| |ε|≤z(3)|4|Lα=↓|41
 9729  .20205|969031|959594|928539|997381|961511|944999|907650|→α_↓
 9729  >| |πln|4|ε|≤f|4|Lα=↓|40.48121|918250|959603|944749|977589|9
 9730  13424|936842|931352|→α_↓>| 1/|πln|4|ε|≤f|4|Lα=↓|42.07808|969
 9731  212|935027|953760|913226|906117|979576|977422|→α_↓>
 9732  | |→α_↓|πln|4ln|42|4|Lα=↓|40.36651|929205|981664|932701|9243
 9732  91|958232|966946|994543|→α_↓>{B2}|J#>{A6}|Ha{U0}{H9L11M29}|π
folio 844 galley 11 "special italic figs in position 4"
 9734  W58320#Computer|4Programming!(Knuth/Addison-Wesley)!Appendix
 9734  !f.|4844!g.|411d|'{A12}{H8L10}|∨T|∨a|∨b|∨l|∨e|7|∨2|;
 9736  {A6}{H9L11}QUANTITIES WHICH ARE FREQUENTLY USED 
 9741  IN STANDARD SUBROUTINES AND IN ANALYSIS OF COMPUTER 
 9749  PROGRAMS, IN |εOCTAL |πNOTATION. THE |εNAME |πOF 
 9756  EACH QUANTITY, APPEARING AT THE LEFT OF THE EQUAL 
 9765  SIGN, IS GIVEN IN DECIMAL NOTATION.|;|J#>!!!!!!0.1|4|∂α=↓|4|
 9772  */|↔0.|↔0|↔6|↔3|↔1|↔4|9|↔6|↔3|↔1|↔4|↔6|9|↔3|↔1|↔4|↔6|↔3|9|↔1|
 9772  ↔4|↔6|↔3|↔1|9|↔4|↔6|↔3|↔1|↔4|9|↔6|↔3|↔1|↔4|↔6|9|↔3|↔1|↔4|↔6|
 9772  ↔3|9|↔1|↔4|↔6|↔3|↔1|9|↔4|↔6|↔3|↔2|'| |\0.01|4|Lα=↓|4|*/|↔0.|↔
 9773  0|↔0|↔5|↔0|↔7|9|↔5|↔3|↔4|↔1|↔2|9|↔1|↔7|↔2|↔7|↔0|9|↔2|↔4|↔3|↔
 9773  6|↔5|9|↔6|↔0|↔5|↔0|↔7|9|↔5|↔3|↔4|↔1|↔2|9|↔1|↔7|↔2|↔7|↔0|9|↔2
 9773  |↔4|↔3|↔6|↔5|9|↔6|↔0|↔5|↔1|\>| 0.001|4|Lα=↓|4|*/|↔0.|↔0|↔0|↔0
 9774  |↔4|↔0|9|↔6|↔1|↔1|↔1|↔5|9|↔6|↔4|↔5|↔7|↔0|9|↔6|↔5|↔1|↔7|↔6|9|
 9774  ↔7|↔6|↔3|↔5|↔5|9|↔4|↔4|↔2|↔6|↔4|9|↔1|↔6|↔2|↔5|↔4|9|↔0|↔2|↔0|
 9774  ↔3|↔0|9|↔4|↔4|↔6|↔7>| |\0.0001|4|Lα=↓|4|*/|↔0.|↔0|↔0|↔0|↔0|↔3
 9775  |9|↔2|↔1|↔5|↔5|↔6|9|↔1|↔3|↔5|↔3|↔0|9|↔7|↔0|↔4|↔1|↔4|9|↔5|↔4|
 9775  ↔5|↔1|↔2|9|↔7|↔5|↔1|↔7|↔0|9|↔3|↔3|↔0|↔2|↔1|9|↔1|↔5|↔0|↔0|↔2|
 9775  9|↔3|↔5|↔2|↔2|\>| 0.00001|4|Lα=↓|4|*/|↔0.|↔0|↔0|↔0|↔0|↔0|9|↔2
 9776  |↔4|↔7|↔6|↔1|9|↔3|↔2|↔6|↔1|↔0|9|↔7|↔0|↔6|↔6|↔4|9|↔3|↔6|↔0|↔4
 9776  |↔1|9|↔0|↔6|↔0|↔7|↔7|9|↔1|↔7|↔4|↔0|↔1|9|↔5|↔6|↔0|↔6|↔3|9|↔3|
 9776  ↔4|↔4|↔2>| |\0.000001|4|Lα=↓|4|*/|↔0.|↔0|↔0|↔0|↔0|↔0|9|↔0|↔2|
 9777  ↔0|↔6|↔1|9|↔5|↔7|↔3|↔6|↔5|9|↔0|↔5|↔5|↔3|↔6|9|↔6|↔6|↔1|↔5|↔1|
 9777  9|↔5|↔5|↔3|↔2|↔3|9|↔0|↔7|↔7|↔4|↔6|9|↔4|↔4|↔4|↔7|↔0|9|↔2|↔6|↔
 9777  0|↔3|\>| 0.0000001|4|Lα=↓|4|*/|↔0.|↔0|↔0|↔0|↔0|↔0|9|↔0|↔0|↔1|
 9778  ↔5|↔3|9|↔2|↔7|↔7|↔4|↔5|9|↔1|↔5|↔2|↔7|↔4|9|↔5|↔3|↔6|↔4|↔4|9|↔
 9778  1|↔2|↔7|↔4|↔1|9|↔7|↔2|↔3|↔1|↔2|9|↔2|↔0|↔3|↔5|↔4|9|↔0|↔2|↔1|↔
 9778  5>| |\0.00000001|4|Lα=↓|4|*/|↔0.|↔0|↔0|↔0|↔0|↔0|9|↔0|↔0|↔0|↔1
 9779  |↔2|9|↔5|↔7|↔1|↔4|↔3|9|↔5|↔6|↔1|↔0|↔6|9|↔0|↔4|↔3|↔0|↔3|9|↔4|
 9779  ↔7|↔3|↔7|↔4|9|↔7|↔7|↔3|↔4|↔1|9|↔0|↔1|↔5|↔1|↔2|9|↔6|↔3|↔3|↔3>
 9780  | |\0.000000001|4|Lα=↓|4|*/|↔0.|↔0|↔0|↔0|↔0|↔0|9|↔0|↔0|↔0|↔0|
 9780  ↔1|9|↔0|↔4|↔5|↔6|↔0|9|↔2|↔7|↔6|↔4|↔0|9|↔4|↔6|↔6|↔5|↔5|9|↔1|↔
 9780  2|↔2|↔6|↔2|9|↔7|↔1|↔4|↔2|↔6|9|↔4|↔0|↔1|↔2|↔4|9|↔2|↔1|↔7|↔4>
 9781  | |\0.0000000001|4|Lα=↓|4|*/|↔0.|↔0|↔0|↔0|↔0|↔0|9|↔0|↔0|↔0|↔0
 9781  |↔0|9|↔0|↔6|↔6|↔7|↔6|9|↔3|↔3|↔7|↔6|↔6|9|↔3|↔5|↔3|↔6|↔7|9|↔5|
 9781  ↔5|↔6|↔5|↔3|9|↔3|↔7|↔2|↔6|↔5|9|↔3|↔4|↔6|↔4|↔2|9|↔0|↔1|↔6|↔3>
 9782  | |\|¬H2|4|Lα=↓|4|*/|↔1.|↔3|↔2|↔4|↔0|↔4|9|↔7|↔4|↔6|↔3|↔1|9|↔7
 9782  |↔7|↔1|↔6|↔7|9|↔4|↔6|↔2|↔2|↔0|9|↔4|↔2|↔6|↔2|↔7|9|↔6|↔6|↔1|↔1
 9782  |↔5|9|↔4|↔6|↔7|↔2|↔5|9|↔1|↔2|↔5|↔7|↔5|9|↔1|↔7|↔4|↔4|\>
 9783  | |¬H3|4|Lα=↓|4|*/|↔1.|↔5|↔6|↔6|↔6|↔3|9|↔6|↔5|↔6|↔4|↔1|9|↔3|↔
 9783  0|↔2|↔3|↔1|9|↔2|↔5|↔1|↔6|↔3|9|↔5|↔4|↔4|↔5|↔3|9|↔5|↔0|↔2|↔6|↔
 9783  5|9|↔6|↔0|↔3|↔6|↔1|9|↔3|↔4|↔0|↔7|↔3|9|↔4|↔2|↔2|↔2|\>
 9784  | |¬H5|4|Lα=↓|4|*/|↔2.|↔1|↔7|↔0|↔6|↔7|9|↔3|↔6|↔3|↔3|↔4|9|↔5|↔
 9784  7|↔7|↔2|↔2|9|↔4|↔7|↔6|↔0|↔2|9|↔5|↔7|↔4|↔7|↔1|9|↔6|↔3|↔0|↔0|↔
 9784  3|9|↔0|↔0|↔5|↔6|↔3|9|↔5|↔5|↔6|↔2|↔0|9|↔3|↔2|↔0|↔2|\>
 9785  | |¬H10|4|Lα=↓|4|*/|↔3.|↔1|↔2|↔3|↔0|↔5|9|↔4|↔0|↔7|↔2|↔6|9|↔6|
 9785  ↔4|↔5|↔5|↔5|9|↔2|↔2|↔4|↔4|↔4|9|↔0|↔2|↔2|↔4|↔2|9|↔5|↔7|↔1|↔0|
 9785  ↔1|9|↔4|↔1|↔4|↔6|↔6|9|↔3|↔3|↔7|↔7|↔5|9|↔2|↔2|↔5|↔3|\>
 9786  | |¬H2|4|Lα=↓|4|*/|↔1.|↔2|↔0|↔5|↔0|↔5|9|↔0|↔5|↔7|↔4|↔6|9|↔1|↔
 9786  5|↔3|↔4|↔5|9|↔0|↔5|↔3|↔4|↔2|9|↔1|↔0|↔7|↔5|↔6|9|↔6|↔5|↔3|↔3|↔
 9786  4|9|↔2|↔5|↔5|↔7|↔4|9|↔2|↔2|↔4|↔1|↔5|9|↔0|↔3|↔0|↔3|\>
 9787  | |¬H3|4|Lα=↓|4|*/|↔1.|↔3|↔4|↔2|↔3|↔3|9|↔5|↔0|↔4|↔4|↔4|9|↔2|↔
 9787  2|↔1|↔7|↔5|9|↔7|↔3|↔1|↔3|↔4|9|↔6|↔7|↔3|↔6|↔3|9|↔7|↔6|↔1|↔3|↔
 9787  3|9|↔0|↔5|↔3|↔3|↔4|9|↔3|↔1|↔1|↔4|↔7|9|↔6|↔0|↔1|↔2|\>
 9788  | |¬H2|4|Lα=↓|4|*/|↔1.|↔1|↔4|↔0|↔6|↔7|9|↔7|↔4|↔0|↔5|↔0|9|↔6|↔
 9788  1|↔5|↔5|↔6|9|↔1|↔2|↔4|↔5|↔5|9|↔7|↔2|↔1|↔5|↔2|9|↔6|↔4|↔4|↔3|↔
 9788  0|9|↔6|↔0|↔2|↔7|↔1|9|↔0|↔2|↔7|↔5|↔5|9|↔7|↔3|↔1|↔4|\>
 9789  | ln|42|4|Lα=↓|4|*/|↔0.|↔5|↔4|↔2|↔7|↔1|9|↔0|↔2|↔7|↔7|↔5|9|↔7|
 9789  ↔5|↔0|↔7|↔1|9|↔7|↔3|↔6|↔3|↔2|9|↔5|↔7|↔1|↔1|↔7|9|↔0|↔7|↔3|↔1|
 9789  ↔6|9|↔3|↔0|↔0|↔0|↔7|9|↔7|↔1|↔3|↔6|↔6|9|↔5|↔3|↔6|↔4|\>
 9790  | ln|43|4|Lα=↓|4|*/|↔1.|↔0|↔6|↔2|↔3|↔7|9|↔2|↔4|↔7|↔5|↔2|9|↔5|
 9790  ↔5|↔0|↔0|↔6|9|↔0|↔5|↔2|↔2|↔7|9|↔3|↔2|↔4|↔4|↔0|9|↔6|↔3|↔0|↔6|
 9790  ↔5|9|↔2|↔5|↔0|↔1|↔2|9|↔3|↔5|↔5|↔7|↔4|9|↔5|↔5|↔3|↔4|\>
 9791  | ln|43|4|Lα=↓|4|*/|↔2.|↔2|↔3|↔2|↔7|↔3|9|↔0|↔6|↔7|↔3|↔5|9|↔5|
 9791  ↔2|↔5|↔2|↔4|9|↔2|↔5|↔4|↔0|↔5|9|↔5|↔6|↔5|↔1|↔2|9|↔6|↔6|↔5|↔4|
 9791  ↔2|9|↔5|↔6|↔0|↔2|↔6|9|↔4|↔6|↔0|↔5|↔0|9|↔5|↔0|↔7|↔1|\>
 9792  | 1/ln|42|4|Lα=↓|4|*/|↔1.|↔3|↔4|↔2|↔5|↔2|9|↔1|↔6|↔6|↔2|↔4|9|↔
 9792  5|↔3|↔4|↔0|↔5|9|↔7|↔7|↔0|↔2|↔7|9|↔3|↔5|↔7|↔5|↔0|9|↔3|↔7|↔7|↔
 9792  6|↔6|9|↔4|↔0|↔6|↔4|↔4|9|↔3|↔5|↔1|↔7|↔5|9|↔0|↔4|↔3|↔5|\>
 9793  | 1/ln|410|4|Lα=↓|4|*/|↔0.|↔3|↔3|↔6|↔2|↔6|9|↔7|↔5|↔4|↔2|↔5|9|
 9793  ↔1|↔1|↔5|↔6|↔2|9|↔4|↔1|↔6|↔1|↔4|9|↔5|↔2|↔3|↔2|↔5|9|↔2|↔7|↔6|
 9793  ↔5|↔5|9|↔1|↔4|↔7|↔5|↔6|9|↔0|↔6|↔2|↔2|\>| |ε|≤p|4|Lα=↓|4|*/|↔3
 9794  .|↔1|↔1|↔0|↔3|↔7|9|↔5|↔5|↔2|↔4|↔2|9|↔1|↔0|↔2|↔6|↔4|9|↔3|↔0|↔
 9794  2|↔1|↔5|9|↔1|↔4|↔2|↔3|↔0|9|↔6|↔3|↔0|↔5|↔0|9|↔5|↔6|↔0|↔0|↔6|9
 9794  |↔7|↔0|↔1|↔6|↔3|9|↔2|↔1|↔1|↔2|\>| 1|¬P|4α=↓|4|≤p/180|4|Lα=↓|
 9795  4|*/|↔0.|↔0|↔1|↔0|↔7|↔3|9|↔7|↔2|↔1|↔5|↔2|9|↔1|↔1|↔2|↔2|↔4|9|↔
 9795  7|↔2|↔3|↔4|↔4|9|↔2|↔5|↔6|↔0|↔3|9|↔5|↔4|↔2|↔7|↔6|9|↔6|↔3|↔3|↔
 9795  5|↔1|9|↔2|↔2|↔0|↔5|↔6|9|↔1|↔1|↔5|↔5|\>| 1/|≤p|4|Lα=↓|4|*/|↔0|
 9796  ↔.|↔2|↔4|↔2|↔7|↔6|9|↔3|↔0|↔1|↔5|↔5|9|↔6|↔2|↔3|↔4|↔4|9|↔2|↔0|
 9796  ↔2|↔5|↔1|9|↔2|↔3|↔7|↔6|↔0|9|↔4|↔7|↔2|↔5|↔7|9|↔5|↔0|↔7|↔6|↔5|
 9796  9|↔1|↔5|↔1|↔5|↔6|9|↔7|↔0|↔0|↔7|\>| |≤p|g2|4|Lα=↓|4|*/|↔1.|↔6|
 9797  ↔7|↔5|↔1|↔7|9|↔1|↔4|↔4|↔6|↔7|9|↔6|↔2|↔1|↔3|↔5|9|↔7|↔1|↔3|↔2|
 9797  ↔2|9|↔2|↔5|↔5|↔6|↔1|9|↔1|↔5|↔4|↔6|↔6|9|↔3|↔0|↔0|↔2|↔1|9|↔4|↔
 9797  0|↔6|↔5|↔4|9|↔3|↔4|↔1|↔0>| |\|¬H|≤p|4α=↓|4|≤)(1/2)|4|Lα=↓|4|
 9798  */|.|↔6|↔1|↔3|↔3|↔7|9|↔6|↔1|↔1|↔0|↔6|9|↔6|↔4|↔7|↔3|↔6|9|↔6|↔5
 9798  |↔2|↔4|↔7|9|↔4|↔7|↔0|↔3|↔5|9|↔4|↔0|↔5|↔1|↔0|9|↔1|↔5|↔2|↔7|↔3
 9798  |9|↔3|↔4|↔4|↔7|↔0|9|↔1|↔7|↔7|↔6|\>| |≤)(1/3)|4|Lα=↓|4|*/|↔2.|
 9799  ↔5|↔3|↔3|↔4|↔7|9|↔3|↔5|↔2|↔3|↔4|9|↔5|↔1|↔0|↔1|↔3|9|↔6|↔1|↔3|
 9799  ↔1|↔6|9|↔7|↔3|↔1|↔0|↔6|9|↔4|↔7|↔6|↔4|↔4|9|↔5|↔4|↔6|↔5|↔3|9|↔
 9799  0|↔0|↔1|↔0|↔6|9|↔6|↔6|↔0|↔5|\>| |≤)(2/3)|4|Lα=↓|4|*/|↔1.|↔2|↔
 9800  6|↔5|↔2|↔3|9|↔5|↔7|↔1|↔1|↔2|9|↔1|↔4|↔1|↔5|↔4|9|↔7|↔4|↔3|↔1|↔
 9800  2|9|↔5|↔4|↔5|↔7|↔2|9|↔3|↔7|↔6|↔5|↔5|9|↔6|↔0|↔1|↔2|↔6|9|↔2|↔3
 9800  |↔2|↔3|↔1|9|↔0|↔2|↔4|↔5|\>| e|4|Lα=↓|4|*/|↔2.|↔5|↔5|↔7|↔6|↔0|
 9801  9|↔5|↔2|↔1|↔3|↔0|9|↔5|↔0|↔5|↔3|↔5|9|↔5|↔1|↔2|↔4|↔6|9|↔5|↔2|↔
 9801  7|↔7|↔3|9|↔4|↔2|↔5|↔4|↔2|9|↔0|↔0|↔4|↔7|↔1|9|↔7|↔2|↔3|↔6|↔3|9
 9801  |↔6|↔1|↔6|↔6|\>| 1/e|4|Lα=↓|4|*/|↔0.|↔2|↔7|↔4|↔2|↔6|9|↔5|↔3|↔
 9802  0|↔6|↔6|9|↔1|↔3|↔1|↔6|↔7|9|↔4|↔6|↔7|↔6|↔1|9|↔5|↔2|↔7|↔2|↔6|9
 9802  |↔7|↔5|↔4|↔3|↔6|9|↔0|↔2|↔4|↔4|↔0|9|↔5|↔2|↔3|↔7|↔1|9|↔0|↔3|↔3
 9802  |↔6|\>| e|g2|4|Lα=↓|4|*/|↔7.|↔3|↔0|↔7|↔1|↔4|9|↔4|↔5|↔6|↔1|↔5|
 9803  9|↔2|↔3|↔3|↔5|↔5|9|↔3|↔3|↔4|↔6|↔0|9|↔6|↔3|↔5|↔0|↔7|9|↔3|↔5|↔
 9803  0|↔4|↔0|9|↔3|↔2|↔6|↔6|↔4|9|↔2|↔5|↔3|↔5|↔6|9|↔5|↔0|↔2|↔2|\>
 9804  | |≤g|4|Lα=↓|4|*/|↔0.|↔4|↔4|↔7|↔4|↔2|9|↔1|↔4|↔7|↔7|↔0|9|↔6|↔7
 9804  |↔6|↔6|↔6|9|↔0|↔6|↔1|↔7|↔2|9|↔2|↔3|↔2|↔1|↔5|9|↔7|↔4|↔3|↔7|↔6
 9804  |9|↔0|↔1|↔0|↔0|↔2|9|↔5|↔1|↔3|↔1|↔3|9|↔2|↔5|↔5|↔2|\>
 9805  | |πln|4|ε|≤p|4|Lα=↓|4|*/|↔1.|↔1|↔1|↔2|↔0|↔6|9|↔4|↔0|↔4|↔4|↔3
 9805  |9|↔4|↔7|↔5|↔0|↔3|9|↔3|↔6|↔4|↔1|↔3|9|↔6|↔5|↔3|↔7|↔4|9|↔5|↔2|
 9805  ↔6|↔6|↔1|9|↔5|↔2|↔4|↔1|↔0|9|↔3|↔7|↔5|↔1|↔1|9|↔4|↔6|↔0|↔6|\>
 9806  | |≤f|4|Lα=↓|4|*/|↔1.|↔4|↔7|↔4|↔3|↔3|9|↔5|↔7|↔1|↔5|↔6|9|↔2|↔7
 9806  |↔7|↔5|↔1|9|↔2|↔3|↔7|↔0|↔1|9|↔2|↔7|↔6|↔3|↔4|9|↔7|↔1|↔4|↔0|↔1
 9806  |9|↔4|↔0|↔2|↔7|↔1|9|↔6|↔6|↔7|↔1|↔0|9|↔1|↔5|↔0|↔1|\>
 9807  | e|g|≤g|4|Lα=↓|4|*/|↔1.|↔6|↔1|↔7|↔7|↔2|9|↔1|↔3|↔4|↔5|↔2|9|↔6
 9807  |↔1|↔1|↔5|↔2|9|↔6|↔5|↔7|↔6|↔1|9|↔2|↔2|↔4|↔7|↔7|9|↔3|↔6|↔5|↔5
 9807  |↔3|9|↔5|↔3|↔3|↔2|↔7|9|↔1|↔7|↔5|↔5|↔4|9|↔2|↔1|↔2|↔6|\>
 9808  | e|g|≤p|g/|g4|4|Lα=↓|4|*/|↔2.|↔1|↔4|↔2|↔7|↔5|9|↔3|↔1|↔5|↔1|↔
 9808  2|9|↔1|↔6|↔1|↔6|↔2|9|↔5|↔2|↔3|↔7|↔0|9|↔3|↔5|↔5|↔3|↔0|9|↔1|↔1
 9808  |↔3|↔4|↔2|9|↔5|↔3|↔5|↔2|↔5|9|↔4|↔4|↔3|↔0|↔7|9|↔0|↔2|↔1|↔7|\>
 9809  | |πsin|41|4|Lα=↓|4|*/|↔0.|↔6|↔5|↔6|↔6|↔5|9|↔2|↔4|↔4|↔3|↔6|9|
 9809  ↔0|↔4|↔4|↔1|↔4|9|↔7|↔3|↔4|↔0|↔2|9|↔0|↔3|↔0|↔6|↔7|9|↔2|↔3|↔6|
 9809  ↔4|↔4|9|↔1|↔1|↔6|↔1|↔2|9|↔0|↔7|↔4|↔7|↔4|9|↔1|↔4|↔5|↔1|\>
 9810  | cos|41|4|Lα=↓|4|*/|↔0.|↔4|↔2|↔4|↔5|↔0|9|↔5|↔0|↔0|↔3|↔7|9|↔3
 9810  |↔2|↔4|↔0|↔6|9|↔4|↔2|↔7|↔1|↔1|9|↔0|↔7|↔0|↔2|↔2|9|↔1|↔4|↔6|↔6
 9810  |↔6|9|↔2|↔7|↔3|↔2|↔0|9|↔7|↔0|↔6|↔7|↔5|9|↔1|↔2|↔3|↔2|\>
 9811  | |ε|≤z(3)|4|Lα=↓|4|*/|↔1.|↔1|↔4|↔7|↔3|↔5|9|↔0|↔0|↔0|↔2|↔3|9|
 9811  ↔6|↔0|↔0|↔1|↔4|9|↔2|↔0|↔4|↔7|↔0|9|↔1|↔5|↔6|↔1|↔3|9|↔4|↔2|↔5|
 9811  ↔6|↔1|9|↔3|↔1|↔7|↔1|↔5|9|↔1|↔0|↔1|↔7|↔7|9|↔0|↔6|↔6|↔2|\>
 9812  | |πln|4|ε|≤f|4|Lα=↓|4|*/|↔0.|↔3|↔6|↔6|↔3|↔0|9|↔2|↔6|↔2|↔5|↔6
 9812  |9|↔6|↔1|↔2|↔1|↔3|9|↔0|↔1|↔1|↔4|↔5|9|↔1|↔3|↔7|↔0|↔0|9|↔4|↔1|
 9812  ↔0|↔0|↔4|9|↔5|↔2|↔2|↔6|↔4|9|↔3|↔0|↔7|↔0|↔0|9|↔4|↔0|↔6|↔5|\>
 9813  | |π1/ln|4|ε|≤f|4|Lα=↓|4|*/|↔2.|↔0|↔4|↔7|↔7|↔6|9|↔6|↔0|↔1|↔1|
 9813  ↔1|9|↔1|↔7|↔1|↔4|↔4|9|↔4|↔1|↔5|↔1|↔2|9|↔1|↔1|↔4|↔3|↔6|9|↔1|↔
 9813  6|↔5|↔7|↔5|9|↔0|↔0|↔3|↔5|↔5|9|↔4|↔3|↔6|↔3|↔0|9|↔4|↔0|↔6|↔5|\
 9813  >| |π|→α_↓ln|4ln|42|4|Lα=↓|4|*/|↔0.|↔2|↔7|↔3|↔5|↔1|9|↔7|↔1|↔2
 9814  |↔3|↔3|9|↔6|↔7|↔2|↔6|↔5|9|↔6|↔3|↔6|↔5|↔0|9|↔1|↔7|↔4|↔0|↔1|9|
 9814  ↔5|↔6|↔6|↔3|↔7|9|↔2|↔6|↔3|↔3|↔4|9|↔3|↔1|↔4|↔5|↔5|9|↔5|↔7|↔0|
 9814  ↔1|\>{B2}|J#>{A6}|↔1|9|↔1|9|↔1|9|↔1|9|↔1|9|↔1|9|↔1|9|↔1|9|↔1
 9816  |9|↔1|9|↔1|9|↔2|9|↔2|9|↔2|9|↔2|9|↔2|9|↔2|9|↔2|9|↔2|9|↔2|9|↔2
 9816  |'{A6}|↔3|9|↔3|9|↔3|9|↔3|9|↔3|9|↔3|9|↔3|9|↔3|9|↔3|9|↔3|9|↔4|
 9817  9|↔4|9|↔4|9|↔4|9|↔4|9|↔4|9|↔4|9|↔4|9|↔4|9|↔4|'
 9818  {A6}|↔5|9|↔5|9|↔5|9|↔5|9|↔5|9|↔5|9|↔5|9|↔5|9|↔5|9|↔5|9|↔6|9|
 9818  ↔6|9|↔6|9|↔6|9|↔6|9|↔6|9|↔6|9|↔6|9|↔6|9|↔6|'{A6}|↔7|9|↔7|9|↔
 9819  7|9|↔7|9|↔7|9|↔7|9|↔7|9|↔7|9|↔7|9|↔7|9|↔8|9|↔8|9|↔8|9|↔8|9|↔
 9819  8|9|↔8|9|↔8|9|↔8|9|↔8|9|↔8|'{A6}|↔9|9|↔9|9|↔9|9|↔9|9|↔9|9|↔9
 9820  |9|↔9|9|↔9|9|↔9|9|↔9|9|↔0|9|↔0|9|↔0|9|↔0|9|↔0|9|↔0|9|↔0|9|↔0
 9820  |9|↔0|9|↔0|'|Hu*?*?*?*?{U0}{H9L11M29}|πW58320#Computer|4Programm
folio 845 galley 12
 9821  ing!(Knuth/Addison-Wesley)!Appendix!f.|4845!g.|412d|'
 9822  {A6}{H10L12}!|9|7For high-precision values of 
 9826  constants not found in this list, see J. Peters, 
 9835  |εTen Place Logarithms of the Numbers from |*/|↔O 
 9843  to |↔O|↔c|↔c|↔c|↔c|↔c|\, |πAppendix to Volume 
 9848  1 ed. by M. Abramowitz and I. A. Stegun (Washington, 
 9858  D.C.: U.S. Govt. Printing O∃ce, 1964), Chapter 
 9865  1.|'!|9|7A table of BErnoulli numbers through 
 9872  B|β2|β5|β0 appears in a paper by D. E. Knuth 
 9881  and T. J. Buckholtz, |εMath. Comp. |π|≡2|≡1 (1967), 
 9889  663<688.|'{A12}{H8L10}|π|∨T|∨a|∨b|∨l|∨e|7|∨3|;
 9891  {A6}{H9L11}VALUES OF HARMONIC NUMBERS, BERNOULLI 
 9896  NUMBERS, AND FIBONACCI NUMBERS FOR SMALL VALUES 
 9903  OF |εn.|;|J#>|∂!!!|∂!!!!!!!!|∂!!!!!!!|∂!!!!!!!|4|4|4|∂!!!!!|
 9906  9|∂!!!!|9|∂!!!|∂|E|;|>n|;H|?|βn|'!!!!!B|βn|'|'
 9913  F|βn|;n|;>{A2}|>|9|10|;0|?|?1|?|9|10|;|9|10|;
 9923  >|>|9|11|;1|?|?|→α_↓1/|?2|'|9|11|;|9|11|;>|>|9|12|;
 9935  3/|?2|'1/|?6|'|9|11|;|9|11|;>|>|9|12|;3/|?2|'
 9946  1/|?6|'|9|11|;|9|12|;>|>|9|13|;11/|?6|'0|?|?|9|12|;
 9958  |9|13|;>|>|9|14|;25/|?12|'|→α_↓1/|?30|'|9|13|;
 9967  |9|14|;>|>|9|15|;137/|?60|'0|?|?|9|15|;|9|15|;
 9977  >|>|9|16|;49/|?20|'1/|?42|'|9|18|;|9|16|;>|>|9|17|;
 9989  363/|?140|'0|?|?13|;|9|17|;>|>|9|18|;761/|?280|'
10000  |→α_↓1/|?30|'21|;|9|18|;>|>|9|19|;7129/|?2520|'
10009  0|?|?34|;|9|19|;>|>10|;7381/|?2520|'5/|?66|'55|;
10021  10|;>|>11|;83711/|?27720|'0|?|?89|;11|;>|>12|;
10034  86021/|?27720|'|→α_↓691/|?2730|'144|9|1|;12|;
10040  >|>13|;1145993/|?360360|'0|?|?233|9|1|;13|;>|>
10051  14|;1171733/|?360360|'7/|?6|'377|9|1|;14|;>|>
10060  15|;1195757/|?360360|'0|?|?610|9|1|;15|;>|>16|;
10070  2436559/|?720720|'|→α_↓3617/|?510|'987|9|1|;16|;
10076  >|>17|;42142223/|?12252240|'0|?|?1597!|2|;17|;
10085  >|>18|;14274301/|?4084080|'43867/|?798|'2584!|2|;
10093  18|;>|>19|;275295799/|?77597520|'0|?|?4181!|2|;
10102  19|;>|>20|;55835135/|?15519504|'|→α_↓174611/|?
10109  330|'6765!|2|;20|;>|>21|;18858053/|?5173168|'
10117  0|?|?10946!|9|3|;21|;>|>22|;19093197/|?5173168|'
10126  854513/|?138|'17711!|9|3|;22|;>|>23|;444316699/|?
10134  118982864|'0|?|?28657!|9|3|;23|;>|>24|;1347822955/|?
10143  356948592|'|→α_↓236364091/|?2730|'46368!|9|3|;
10147  24|;>|>25|;34052522467/|?8923714800|'0|?|?75025!|9|3|;
10156  25|;>{B2}|J#>{A3}{H10L12}|πFor any |εx, |πlet 
10163  |εH|βx|4α=↓|4|↔k|uc|)n|¬R1|)|1|1|↔a|(1|d2n|)|4α_↓|4|(1|d2n|4
10163  α+↓|4x|)|↔s. |πThen|'{A9}|ε|hH|β1|β/|β5|4|∂α=↓|46|4α_↓|4|9|≤
10165  p|≤f|4|↔H2|4α+↓|4|≤f|4α_↓|4|9(3|4α_↓|4|≤f)|πln|45|4α_↓|4(|ε|
10165  ≤f|4α_↓|4|9)|πln(2|4α+↓|4|ε|≤f),|E|n|;| H|β1|β/|β2|4|Lα=↓|42
10166  |4α_↓|42|4|πln|42,>{A4}| |εH|β1|β/|β3|4|Lα=↓|43|4α_↓|4|f1|d3
10167  2|)|≤p/|¬H|v43|)|4α_↓|4|f3|d32|)|4|πln|43,>{A4}| |εH|β2|β/|β
10168  3|4|Lα=↓|4|f3|d32|)|4α+↓|4|f1|d32|)|≤p/|¬H|v43|)|4α_↓|4|f3|d
10168  32|)|4|πln|43,>{A4}|ε| H|β1|β/|β4|4|Lα=↓|44|4α_↓|4|f1|d32|)|
10169  ≤p|4α_↓|43|4|πln|42,>{A4}| |εH|β3|β/|β4|4|Lα=↓|4|f4|d33|)|4α
10170  +↓|4|f1|d32|)|≤p|4α_↓|43|4|πln|42,>| |εH|β1|β/|β5|4|Lα=↓|45|
10171  4α_↓|4|f1|d32|)|≤p|≤f|4|↔H|v2|(2|4α+↓|4|≤f|d25|)|)|4α_↓|4|f1
10171  |d32|)(3|4α_↓|4|≤f)|πln|45|4α_↓|4(|ε|≤f|4α_↓|4|f1|d32|))|πln
10171  (2|4α+↓|4|ε|≤f),>{A4}| H|β2|β/|β5|4|Lα=↓|4|f5|d32|)|4α_↓|4|f
10172  1|d32|)|≤p/|≤f|¬H|v42|4α+↓|4|≤f|)|4α_↓|4|f1|d32|)(2|4α+↓|4|≤
10172  f)|πln|45|4α+↓|4(|ε|≤f|4α_↓|4|f1|d32|))|πln(2|4α+↓|4|ε|≤f),>
10173  {A4}| H|β3|β/|β5|4|Lα=↓|4|f5|d33|)|4α+↓|4|f1|d32|)|≤p/|≤f|¬H
10173  |v42|4α+↓|4|≤f|)|4α_↓|4|f1|d32|)(2|4α+↓|4|≤f)|πln|45|4α+↓|4(
10173  |ε|≤f|4α_↓|4|f1|d32|))|πln(2|4α+↓|4|ε|≤f),>| H|β4|β/|β5|4|Lα
10174  =↓|4|f5|d34|)|4α+↓|4|f1|d32|)|≤p|≤f|4|↔H|v2|(2|4α+↓|4|≤f|d25
10174  |)|)|4α_↓|4|f1|d32|)(3|4α_↓|4|≤f)|πln|45|4α_↓|4(|ε|≤f|4α_↓|4
10174  |f1|d32|)|πln(2|4α+↓|4|ε|≤f),>{A4}| H|β1|β/|β6|4α=↓|46|4α_↓|
10175  4|f1|d32|)|≤p|¬H|v43|)|4α_↓|4|π2|4ln|42|4α_↓|4|f3|d32|)|4ln|
10175  43,>{A4}| |εH|β5|β/|β6|4|Lα=↓|4|f6|d35|)|4α+↓|4|f1|d32|)|≤p|
10176  ¬H|v43|)|4α_↓|42|4|πln|42|4α_↓|4|f3|d32|)|4ln|43,>
10177  {A9}and, in general, when |ε0|4|¬W|4p|4|¬W|4q 
10182  (|πcf. exercise 1.2.9<19),|'{A9}|εH|βp|β/|βq|4α=↓|4|(q|d2p|)
10185  |4α_↓|4|f1|d32|)|≤p|4|πcot|4|(|εp|d2q|)|4|≤p|4α_↓|4|πln|4|ε2
10185  q|4α+↓|42|1|1|↔k|uc|)1|¬En|¬Wq/2|)|1|1|πcos|4|(|ε2|≤pnp|d2q|
10185  )|4|πln|4sin|4|(|εn|d2q|)|4|≤p.|;{A24}{H11}|π|!A|1|1|!P|1|1|
10186  !P|1|1|!E|1|1|!N|1|1|!D|1|1|!I|1|1|!X|4|4|4|!B|'
10187  {A69}{H20L20}|∨I|∨N|∨D|∨E|∨X|9|∨T|∨O|9|∨N|∨O|∨T|∨A|∨T|∨I|∨O|
10187  ∨N|∨S|?{A51}{H10L12}|πIn the following formulas, 
10192  letters which are not further quali_ed have the 
10200  following signi_cance:|'{A9}|h|εx, y, z!|∂|πnonnegative 
10205  integer-valued arithmetic expression|E|n|;| |εj, 
10209  k!|L|πinteger-valued arithmetic expression>{A4}|ε| m, 
10213  n!|L|πnonnegative integer-valued arithmetic expression>
10217  {A4}| |εx, y, z!|L|πreal-valued arithmetic expression>
10222  {A4}| |εf!|L|πreal-valued function>{A12}|∂!!!!!!!!!|9|5|∂!!!
10224  !!!!!!!!!!!!!!!!!|∂!!!!!|∂|E|;|>|;|;|≡S|≡e|≡c|≡t|≡i|≡o|≡n|;
10229  >|≡F|≡o|≡r|≡m|≡a|≡l|7|≡s|≡y|≡m|≡b|≡o|≡l|≡i|≡s|≡m!|;
10231  |≡M|≡e|≡a|≡n|≡i|≡n|≡g|;|≡r|≡e|≡f|≡e|≡r|≡e|≡n|≡c|≡e|;
10233  >{B2}|J#>|>|εA|βn!|?|πthe |εn|πth element of 
10241  linear array |εA|'>{A3}|>A|βm|βn!|?|πthe element 
10249  in row |εm, |πcolumn |εn |πof rectan-|'>|>|;gular 
10260  array |εA|'>{A3}|>A[n]!|?|πequivalent to |εA|βn|'
10268  |91.1|'>{A3}|>A[m, n]!|?|πequivalent to |εA|βm|βn|'
10276  |91.1|'>{A3}|>V|4|¬L|4E!|?|πgive variable |εV 
10283  |πthe value of expression |εE|'|91.1|'>{A3}|>
10291  U|4|≠l|4V!|?|πinterchange the values of variables 
10297  |εU |πand |εV|'|91.1|'>{A3}(B|4|*2]|4E|β1; E|β2)!|?
10304  |πconditional expression: denotes |εE|β1 |πif 
10309  |εB |πis |'>|>|;true, if |εB |πis false|'>{A3}|>
10322  |ε|≤d|βj|βk!|?|πKronecker delta: (|εj|4α=↓|4h|4|*2]|41; 
10326  0)|'|91.2.6|'>{A3}|>|↔k|uc|)R(k)|)|4f(k)!|?>{B26}|>
10333  |;|πsum of all |εf(k) |πsuch that |εk |πis an 
10343  integer and|'>|>|;relation |εR(k) |πis true|'
10352  |91.2.3|'>{A4}|>|ε|uw|≥u|uc|)R(k)|)|4f(k)!|?>
10357  {B25}|>|;|πproduct of all |εf(k) |πsuch that 
10365  |εk |πis an integer|'>|>|;and relation |εR(k) 
10375  |πis true|'|91.2.3|'>{A4}|>|uwmin|uc|)|.|εR(k)|)|4f(k)!|?
10381  >{B25}|>|;|πminimum value of all |εf(k) |πsuch 
10390  that |εk |πis an|'>|>|;integer and relation |εR(k) 
10401  |πis true|'|91.2.3|'>{A4}|>|uwmax|uc|)|.|εR(k)|)|4f(k)!|?
10407  >{B25}|>|;|πmaximum value of all |εf(k) |πsuch 
10416  that |εk |πis an|'>|>|;integer and relation |εR(k) 
10427  |πis true|'|91.2.3|'>{A4}|>|εj|¬Dk!|?j |πdivides 
10435  |εk: k|4|πmod|4|εj|4α=↓|40|'|91.2.4|'>{A3}|>|πgcd(|εj, 
10441  k)!|?|πgreatest common divisor of |εj |πand |εk:|'
10449  >|>|;(j|4α=↓|4k|4α=↓|40|4|*2]|40;!!|π|uwmax|uc|)|ε|.d|¬Dj,d|¬
10452  Dk|)|4d)|;|94.5.2|'>{A6}|>|πlcm(|εj, k)!|?|πleast 
10459  common multiple of |εj |πand |εk:|'>|>|;(j|4α=↓|4k|4α=↓|40|4
10468  |*2]|40;!!|π|uwmin|uc|)|ε|.d|¬Q0|dj|¬Dd,k|¬Dd|)|;
10469  |94.5.2|'>{A6}|>|πdet(|εA)!|?|πdeterminant of 
10475  square matrix |εA|'|91.2.3|'>{A3}|>A|gT!|?|πtranspose 
10483  of rectangular array |εA:|'>{A6}|>|;A|gT[j, k]|4α=↓|4A[k, 
10492  j]|;|91.2.3|'>{A6}|>x|gy!|?x |πto the |εy |πpower, 
10502  |εx |πpositive|'|91.2.2|'>{A3}|>|εx|gk!|?x |πto 
10510  the |εk|πth power:|'{A6}|>|;|↔a|εk|4|¬R|40|4|*2]|4|uw|≥u|uc|)
10515  0|¬Ej|¬Wk|)|4x;!!1/x|gα_↓|gk|↔s|;|91.2.2|'>{A6}|>
10519  x|=|g3|gk!|?x |πupper |εk:|'>{A6}|>|;|↔a|ur|)|)k|4|¬R|40|4|*2
10526  ]|4x(x|4α+↓|41)|4.|4.|4.|4(x|4α+↓|4k|4α_↓|41)|'
10527  >{B6}|>|;α=↓|uw|≥u|uc|)0|¬Ej|¬Wk|)|4(x|4α+↓|4j);!!1/(x|4α+↓|
10530  4k)|gα_↓|gk|↔s|9|4|?|91.2.6|'>{A6}|>|=|β3|gk!|?
10535  x |πlower |εk:|'>{A6}|>|;|↔a|ur|)|)k|4|¬R|40|4|*2]|4x(x|4α_↓|
10541  41)|4.|4.|4.|4(x|4α_↓|4k|4α+↓|41)|'>{B6}|>|;α=↓|4|uw|≥u|uc|)
10545  0|¬Ej|¬Wk|)|4(x|4α_↓|4j);|9|4|;>{A6}|>|;1/(x|4α_↓|4k)|uw#|uc
10549  |,α_↓k|)|)|↔s|ur|)|)|3α=↓|4(|→α_↓1)|gk(|→α_↓x)|=|g∩|gk|9|4|?
10550  1.2.6|'>{A6}|>n*3!|?n |πfactorial: 1|4αo↓|42|4αo↓|4.|4.|4.|4α
10556  o↓|4|εn|4α=↓|4n|=|β3|gn|'|91.2.5|'>{A3}|>|↔a|(x|d5k|)|↔s!|?
10561  |πbinomial coe∃cient: (|εk|4|¬W|40|4|*2]|40; x|=|β3|gk/k*3)|'
10565  |91.2.6|'{A3}|↔a|(n|d5n|β1,|4n|β2,|4.|4.|4.|4,|4n|βm|)|↔s!|?
10567  >{B26}|>|;|πmultinomial coe∃cient,|'{A2}|>|;|εn|4α=↓|4n|β1|4
10574  α+↓|4n|β2|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|4n|βm|;1.2.6|'
10576  >{A3}|>|↔d|(n|d5m|)|↔f!|?|πStirling number of 
10582  _rst kind:|'>|>|;|ε|uw|↔k|uc|)0|¬Wk|β1|¬Wk|β2|¬W|1|¬O|1|¬O|1
10587  |¬O|1|¬Wk|βn|βα_↓|βm|¬Wn|)|4k|β1k|β2|4.|4.|4.|4k|βn|βα_↓|βm|
10587  ;|91.2.6|'>{A3}|>|↔A|(n|d5m|)|↔S!|?|πStprling 
10593  number of second kind:|'>|>|;|ε|↔k|uc|)1|¬Ek|β1|¬Ek|β2|¬E|1|
10600  ¬O|1|¬O|1|¬O|1|¬Ek|βn|βα_↓|βm|¬Em|)|4k|β1k|β2|4.|4.|4.|4k|βn
10600  |βα_↓|βm|;|91.2.6|'>{A6}|>|"Cx|β1, x|β2,|4.|4.|4.|4,|4x|βn|"
10605  C!|?|πcontinued fraction:|'>{A2}|ε1/{H12}({H10}x|β1|4α+↓|41/
10609  (x|β2|4α+↓|41/(αo↓|4αo↓|4αo↓|4α+↓|41/(x|βn)|4.|4.|4.)){H12})
10609  {H10}|;|94.5.3|'>{A3}|>X|4|¬O|4Y!|?|πdot product 
10616  of vectors |εX|4α=↓|4(x|β1,|4.|4.|4.|4,|4x|βn) 
10619  |πand|'>|>|;|εY|4α=↓|4(y|β1,|4.|4.|4.|4,|4y|βn): 
10624  x|β1y|β1|4α+↓|4αo↓|4αo↓|4αo↓|4α+↓|4x|βny|βn|'
10625  >{A3}|>(.|4.|4.|4a|β1a|β0|4|¬O|4a|βα_↓|β1|4.|4.|4.)|βb!|?
10628  |πradix-|εb |πpositional notation: |ε|¬K|βk|4a|βkb|gk|'
10632  |94.1|'>{A3}|>|¬Ta|4|¬G|4R(a)|¬Y!|?|πset of all 
10639  |εa |πfor which the relation |εR(a) |πis true|'
10647  >{A3}|>|ε|¬Ta|β1,|4.|4.|4.|4,|4a|βn|¬Y!|?|πthe 
10651  set |¬T|εa|βk|4|¬G|41|4|¬E|4k|4|¬E|4n|¬Y|'>{A3}|>
10655  |¬Tx|¬Y!|?|πin contexts where a real value, not 
10663  a set, is|'>|>|;required, denotes fractional 
10672  part: |εx|4|πmod|41|'|93.3.3|'>{A3}|>|ε[y, z)!|?
10679  |πhalf-open interval: |ε|¬Tx|4|¬G|4y|4|¬E|4x|4|¬W|4z|¬Y|'
10682  >{A3}|>|¬FS|¬F!|?|πcardinal: number of elements 
10689  in set |εS|'>{A3}|>{H12}({H10}(x){H12}){H10}!|?
10695  |πsawtooth function|'|93.3.3|'>{A3}|>|ε|¬Gx|¬G!|?
10701  |πabsolute value of |εx: (x|4|¬W|40|4|"M|4|→α_↓x; 
10706  x)|'>{B3}|>|"lx|"L!|?|π⊗oor of |εx, |πgreatest 
10714  integer function: |uw|πmax|)|ε|.k|¬Ex|)|4k|'|91.2.4|'
10718  >{A3}|>x|4|πmod|4|εy!|?|πmod function: (|εy|4α=↓|40|4|"M|4x;
10723   x|4α_↓|4y|"lx/y|"L)|'|91.2.4|'|Hu*?*?*?{U0}{H9L11M29}|πW58320#
10726  Computer|4Programming!(Knuth/Addison-Wesley)!Appendix!f.|484
folio 849 galley 13
10726  9!g.|413d|'{A6}{H10L12}|∂!!!!!!!!!|9|5|∂!!!!!!!!!!!!!!!!!!!!
10727  |∂!!!!!|∂|E|;|>|;|;|≡S|≡e|≡c|≡t|≡i|≡o|≡n|;>|>
10734  |≡F|≡o|≡r|≡m|≡a|≡l|7|≡s|≡y|≡m|≡b|≡o|≡l|≡i|≡s|≡m|9|9|;
10735  |≡M|≡e|≡a|≡n|≡i|≡n|≡g|;|≡r|≡e|≡f|≡e|≡r|≡e|≡n|≡c|≡e|;
10737  >{B2}|J#>{A6}|>|;|;|≡S|≡e|≡c|≡t|≡i|≡o|≡n|;>|>
10745  |≡F|≡o|≡r|≡m|≡a|≡l|7|≡s|≡y|≡m|≡b|≡o|≡l|≡i|≡s|≡m|9|9|;
10746  |≡M|≡e|≡a|≡n|≡i|≡n|≡g|;|≡r|≡e|≡f|≡e|≡r|≡e|≡n|≡c|≡e|;
10748  >{B2}|J#>{A6}|>|;|;|≡S|≡e|≡c|≡t|≡i|≡o|≡n|;>|>
10756  |≡F|≡o|≡r|≡m|≡a|≡l|7|≡s|≡y|≡m|≡b|≡o|≡l|≡i|≡s|≡m|9|9|;
10757  |≡M|≡e|≡a|≡n|≡i|≡n|≡g|;|≡r|≡e|≡f|≡e|≡r|≡e|≡n|≡c|≡e|;
10759  >{B2}|J#>{A6}|>|;|;|≡S|≡e|≡c|≡t|≡i|≡o|≡n|;>|>
10767  |≡F|≡o|≡r|≡m|≡a|≡l|7|≡s|≡y|≡m|≡b|≡o|≡l|≡i|≡s|≡m|9|9|;
10768  |≡M|≡e|≡a|≡n|≡i|≡n|≡g|;|≡r|≡e|≡f|≡e|≡r|≡e|≡n|≡c|≡e|;
10770  >{B2}|J#>|>|εu(x)|πmod|4|εv(x)!|?|πremainder 
10775  of polynomial |εu(x) |πdivided by |εv(x)|'|94.6.1|'
10782  >{A3}|>x|4|"o|4y (|πmodulo |εz)!|?|πrelation 
10788  of congruence: |εx|4|πmod|4|εz|4α=↓|4y|4|πmod|4|εz|'
10791  |91.2.4|'>{A3}|>|πlog|ε|βb|4x!|?|πlogarithm, 
10796  base |εb, |πof |εx (|πreal positive |εb|4|=|↔6α=↓|41):|'
10803  >|>|;x|4α=↓|4b|π|gl|go|gg|ε|rb|1|1|gx|'|91.2.2|'
10808  >{A3}|>|πlg|4|εx!|?|πbinary logarithm: log|β2|4|εx|'
10814  |91.2.2|'>{A3}|>|πln|4|εx!|?|πnatural logarithm: 
10820  log|ε|βe|4x|'|91.2.2|'>{A3}|>|πexp|4|εx!|?|πexponential 
10826  of |εx: e|gx|'|91.2.2|'>{A3}|>|↔bX|βn|↔v!|?|πthe 
10834  in_nite sequence |εX|β0, X|β1, X|β2,|4.|4.|4. 
10839  (|πhere |εn|'>|>|;|πis a letter which is part 
10850  of the symbol)|'|91.2.9|'>{A3}|>|εf|¬S(x)!|?|πderivative 
10858  of |εf |πat |εx|'|91.2.9|'>{A3}|>f|¬C(x)!|?|πsecond 
10867  derivative of |εf |πat |εx|'|91.2.10|'>{A3}|>
10875  f|g(|gn|g)(x)!|?n|πth derivative: |ε{H12}({H10}n|4α=↓|40|4|"
10878  M|4f(x);|'>{A3}|>|;g|¬S(x)!!|πwhere!!|εg(x)|4α=↓|4f|g(|gn|gα
10882  _↓|g1|g)(x){H12}){H10}|'|91.2.11.2|'>{B3}|>H|ur(x)|)n|)!|?
10887  1|4α+↓|41/2|gx|4α+↓|4|¬O|4|¬O|4|¬O|4α+↓|41/n|gx|4α=↓|4|↔k|uc
10887  |)1|¬Ek|¬En|)|41/k|gx|'|91.2.7|'>{A3}|>H|βn!|?
10892  |πharmonic number: |εH|ur(1)|)n|)|'|91.2.7|'>
10897  {A3}|>F|βn!|?|πFibonacci number:|'>|>|;{A3}(|εn|4|¬E|41|4|"M
10904  |4n;|7F|βn|βα_↓|β1|4α+↓|4F|βn|βα_↓|β2)|;|91.2.8|'
10906  >{A3}|>B|βn!|?|πBernoulli number|'|91.2.11.2|'
10912  >{A3}|>B(|εx, y)!|?|πBeta function|'|91.2.6|'
10919  >{A3}|>sign(|εx)!|?|πsign of |εx:|'>{A3}|>|;{H12}({H10}x|4α=
10928  ↓|40|4|"M|40;|7(x|4|¬Q|40|4|"M|4|→α+↓1;|7|→α_↓1){H12}){H10}|
10928  ;>{A3}|>|≤d(x)!|?|πcharacteristic function of 
10935  the integers|'|93.3.3|'>{A3}|ε|>|≤z(x)!|?|πzeta 
10942  function: |εH|ur(x)|)|¬X|) |πwhen |εx|4|¬Q|41|'
10946  |91.2.7|'>{A3}|>|≤)(x)!|?|πgamma function: |ε|≤g(x,|4|¬X); 
10953  (x|4α_↓|41)*3 |πwhen |εx|'>|>|;|πis a positive 
10962  integer|'|91.2.5|'>{A3}|>|ε|≤g(x, y)!|?|πincomplete 
10969  gamma function|'|91.2.11.3|'>{A3}|>|ε|≤g!|?|πEuler's 
10976  constant|'|91.2.7|'>{B3}|>|εe!|?|πbase of natural 
10984  logarithms: |ε|↔k|uc|)k|¬R0|)|41/k*3|'|91.2.2|'
10987  >{A3}|>|→|¬X!|?|πin_nity: larger than any number|'
10995  >{A3}|>|=/0!|?empty set (set with no elements)|'
11004  >{A3}|>|ε|≤f!|?|πgolden ratio, |f1|d32|)(1|4α+↓|4|¬H|v45|))|
11009  '|91.2.8|'>{B3}|>|ε|≤'(n)!|?|πEuler's totient 
11016  function: |ε|↔k|uc|)0|¬Ek|¬Wn|d|πgcd(|εk,n)α=↓1|)|1|11|'
11018  |91.2.4|'>{A3}|>|≤m(n)!|?|πM|=4obius function|'
11024  |94.5.2|'>{A3}|>|ε|≤!(n)!|?|πvon Mangoldt's function|'
11031  |94.5.3|'>{A3}|>|εp(n)!|?|πnumber of partitions 
11038  of |εn|'|91.2.1|'>{A3}|>|πPr{H12}({H10}|εS(n){H12}){H10}!|?
11044  |πprobability that statement |εS(n) |πis true, 
11050  for|'>|>|;``random'' |εn|'|93.5,|44.2.4|'>{A3}|>
11059  O{H12}({H10}f(n){H12}){H10}!|?|πbig-oh of |εf(n) 
11063  |πas |εn|4|¬M|4|¬X|'|91.2.11.1|'>{A3}|>O{H12}({H10}f(x){H12}
11068  ){H10}!|?|πbig-oh of |εf(x), |πfor small |εx 
11075  |π(or for |εx |πin some|'>|>|;speci_ed range)|'
11085  |91.2.11.1|'>{A3}|>!(min|4|εx|β1,|4|πave|4|εx|β2,!|'
11089  |πa random variable having minimum value |εx|β1,|'
11096  >|>|πmax|4|εx|β3,|4dev|4x|β4)!|?|πaverage (``expected'') 
11101  value |εx|β2, |πmaximum|'>|>|;value |εx|β3, |πstandard 
11110  deviation |εx|β4|'|91.2.10|'>{A3}|>|πmean(|εg)!|?
11116  |πmean value of probability distribution repre-|'
11122  >|>|;sented by generating function |εg: g|¬S(1)|'
11131  |91.2.10|'>{A3}|>|πvar(|εg)!|?|πvariance of probability 
11138  distribution repre-|'>|>|;sented by generating 
11146  function |εg:|'>{A6}|>|;g|¬C(1)|4α+↓|4|¬S(1)|4α_↓|4g|¬S(1)|g
11151  2|;|91.2.10|'>{A6}|>|πdeg(|εu)!|?|πdegree of 
11158  polynomial |εu|'|94.6|'>{A3}|>|λ3(u)!|?|πleading 
11165  coe∃cient of polynomial |εu|'|94.6|'>{A3}|>|πcont(|εu)!|?
11173  |πcontent of polynomial |εu|'|94.6.1|'>{A3}|>
11180  |πpp{H12}({H10}|εu(x){H12}){H10}!|?|πprimitive|9part|9of|9po
11181  lynomial|9|εu,|9|πevaluated|'>|>|;at |εx|'|94.6.1|'
11188  >{A3}|>|λR(w)!|?|πreal part of complex number 
11196  |εw|'>{A3}|>|λI(w)!|?|πimaginary part of complex 
11204  number |εw|'>{A3}|>|=3w!|?|πcomplex conjugate: 
11211  |ε|λR(w)|4α_↓|4|λI(w)|'>{A3}|>|;|πend of algorithm, 
11218  program, or proof|'|91.1|'>{A3}|>|¬G|4|4|¬G!|?
11225  one blank space|'|9A.1|'>{A3}|>rA!|?register 
11233  A (accumulator) of |¬m|¬i|¬x|'|9A.1|'>{A3}|>rX!|?
11241  register X (extension) of |¬m|¬i|¬x|'|9A.1|'>
11248  {A3}|>rI1,|4.|4.|4.|4,|4rI6!|?(index) registers 
11252  I1,|4.|4.|4.|4,|4I6 of |¬m|¬i|¬x|'|9A.1|'>{A3}|>
11258  rJ!|?(jump) register J of |¬m|¬i|¬x|'|9A.1|'>
11266  {A3}|>|≤#|¬l|¬.|¬r|≤&!|?partial _eld of |¬m|¬i|¬x 
11272  word, 0|4|¬E|4|¬l|4|¬E|4|¬r|4|¬E|45|'|9A.1|'>
11276  {A3}|>|¬o|¬p|9|9|¬a|¬d|¬d|¬r|¬e|¬s|¬s|¬,|¬i|≤#|¬f|≤&!|?
11278  notation for |¬m|¬i|¬x instruction|'|9A.1,|4A.2|'
11283  >{A3}|>|εu!|?|πunit of time in |¬m|¬i|¬x|'|9A.1|'
11292  >{A3}|>{J3}|≤⊂!|?``self'' in |¬m|¬i|¬x|¬a|¬l|'
11298  |9A.2|'>{A3}|>|¬0|¬f|¬,|4|¬1|¬f|¬,|4|¬2|¬f|¬,|4.|4.|4.|4|¬,|
11301  4|¬9|¬f!|?``forward'' local symbol in |¬m|¬i|¬x|¬a|¬l|'
11307  |9A.2|'>{A3}|>|¬0|¬b|¬,|4|¬1|¬b|¬,|4|¬2|¬b|¬,|4.|4.|4.|4|¬,|
11310  4|¬9|¬b!|?``backward'' local symbol in |¬m|¬i|¬x|¬a|¬l|'
11316  |9A.2|'>{A3}|>|¬0|¬h|¬,|4|¬1|¬h|¬,|4|¬2|¬h|¬,|4.|4.|4.|4|≡,|
11319  4|¬9|¬h!|?``here'' local symbol in |¬m|¬i|¬x|¬a|¬l|'
11325  |9A.2|'>|Hu*?*?*?*?*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!*!
11327