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Chapter 1.
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Chapter 2.
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[]\T1/ptm/m/n/10.95 If you are run-ning Mac OS, you can use the Finder to go to
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\T1/ptm/m/n/10.95 TeXmacs ([][]$\T1/pcr/m/n/10.95 http : / / www . texmacs . or
g$[][]\T1/ptm/m/n/10.95 ) is a so-phis-ti-cated word pro-ces-sor with
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Chapter 3.
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\T1/pcr/m/n/10.95 Cas configuration
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[]\T1/pcr/m/n/10.95 xyztrange(\T1/ptm/m/it/10.95 x-,x+,y-,y+,z-,z+,t-,t+,wx-,wx
+,wy-,wy+,show_axes,class_min,class_size\T1/pcr/m/n/10.95 )
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\T1/ptm/m/n/10.95 menu item will save a spread-sheet in a comma-separated val-u
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ave
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\T1/pcr/m/n/10.95 Prg$\U/msa/m/n/10.95 I$\T1/pcr/m/n/10.95 New program \T1/ptm/
m/n/10.95 or with \T1/pcr/m/n/10.95 Alt-P\T1/ptm/m/n/10.95 . If you se-lect thi
s item, you will be prompted
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[36]
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[]\T1/ptm/m/n/10.95 If you pre-fer to save the printed form as a file, you can 
use the \T1/pcr/m/n/10.95 File$\U/msa/m/n/10.95 I$\T1/pcr/m/n/10.95 Print$\U/ms
a/m/n/10.95 I$\T1/pcr/m/n/10.95 Preview
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ph2tex graph3d2tex
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\T1/ptm/m/n/10.95 menu; the \T1/pcr/m/n/10.95 Print$\U/msa/m/n/10.95 I$\T1/pcr/
m/n/10.95 LaTeX print \T1/ptm/m/n/10.95 will give you sev-eral files \T1/pcr/m/
n/10.95 DispG.tex\T1/ptm/m/n/10.95 , \T1/pcr/m/n/10.95 DispG.dvi\T1/ptm/m/n/10.
95 ,
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c>[] 
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Chapter 4.
[43

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[]\T1/ptm/m/n/10.95 To add a com-ment to a pro-gram, rather than a ses-sion, yo
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\T1/ptm/m/n/10.95 list of pre-vi-ous com-mands and dis-play them on the cur-ren
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\T1/ptm/m/n/10.95 The con-fig-u-ra-tion screen can be re-opened with the \T1/pc
r/m/n/10.95 Edit$\U/msa/m/n/10.95 I$\T1/pcr/m/n/10.95 Configuration$\U/msa/m/n/
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Chapter 5.
[51]
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[]\T1/ptm/b/n/14.4 Symbolic con-stants: \T1/pcr/b/n/14.4 e pi infinity inf i eu
ler_gamma
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[]\T1/ptm/m/n/10.95 Euler's con-stant $\OML/cmm/m/it/10.95 ^^M$\T1/ptm/m/n/10.9
5 ; namely,
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[52]
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\T1/ptm/m/n/10.95 Sec-tion [][]6.39.2[][] p.[][]399[][] for in-for-ma-tion on \
T1/pcr/m/n/10.95 seq($[]$)\T1/ptm/m/n/10.95 ). For ex-am-ple, \T1/pcr/m/n/10.95
 seq([0,2])=(0,0)
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[53] [54] [55] [56]
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\T1/ptm/m/n/10.95 (see Sec-tion [][]6.39[][] p.[][]398[][]). Sup-pose you give 
the vari-able \T1/pcr/m/n/10.95 a \T1/ptm/m/n/10.95 the value \T1/pcr/m/n/10.95
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[57]
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[]\T1/ptm/b/n/12 Storing and re-call-ing vari-ables and their val-ues: \T1/pcr/
b/n/12 archive unarchive
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[58]
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[]\T1/ptm/m/n/10.95 The \T1/pcr/m/n/10.95 unarchive \T1/ptm/m/n/10.95 com-mand 
will read the val-ues from a file cre-ated with \T1/pcr/m/n/10.95 archive\T1/pt
m/m/n/10.95 . 
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\T1/ptm/m/n/10.95 You can also as-sume a vari-able is in one of the do-mains \T
1/pcr/m/n/10.95 real\T1/ptm/m/n/10.95 , \T1/pcr/m/n/10.95 integer\T1/ptm/m/n/10
.95 , \T1/pcr/m/n/10.95 complex
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[]\T1/pcr/m/n/10.95 NewFold(\T1/ptm/m/it/10.95 MyIntDir) \T1/ptm/m/n/10.95 cre-
ates a new in-ter-nal di-rec-tory named \T1/ptm/m/it/10.95 My-Int-Dir\T1/ptm/m/
n/10.95 . (Note
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[]\T1/pcr/m/n/10.95 SetFold(\T1/ptm/m/it/10.95 MyIntDir\T1/pcr/m/n/10.95 ) \T1/
ptm/m/n/10.95 makes \T1/ptm/m/it/10.95 My-Int-Dir \T1/ptm/m/n/10.95 the work-in
g di-rec-tory (see Sec-tion [][]5.6.1[][]
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[]\T1/ptm/m/n/10.95 Finally, you can print out the in-ter-nal di-rec-tory that 
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Chapter 6.
[69

] [70]
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[]\T1/pcr/m/n/10.95 when(\T1/ptm/m/it/10.95 condition, true-result, false-resul
t\T1/pcr/m/n/10.95 ) \T1/ptm/m/n/10.95 (and \T1/pcr/m/n/10.95 IFTE(\T1/ptm/m/it
/10.95 condition, true-result,
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[71]
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[]\T1/pcr/m/n/10.95 Xcas \T1/ptm/m/n/10.95 has the stan-dard boolean op-er-a-to
rs, as fol-lows ($\OML/cmm/m/it/10.95 a$ \T1/ptm/m/n/10.95 and $\OML/cmm/m/it/1
0.95 b$ \T1/ptm/m/n/10.95 are two booleans): 
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\T1/ptm/m/n/10.95 These are in-fixed op-er-a-tors. \T1/pcr/m/n/10.95 ($\OML/cmm
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\T1/ptm/m/n/10.95 (or \T1/pcr/m/n/10.95 ($\OML/cmm/m/it/10.95 a$ \T1/pcr/m/n/10
.95 || $\OML/cmm/m/it/10.95 b$\T1/pcr/m/n/10.95 )\T1/ptm/m/n/10.95 ) re-turns \
T1/pcr/m/n/10.95 0 \T1/ptm/m/n/10.95 (or \T1/pcr/m/n/10.95 false\T1/ptm/m/n/10.
95 )
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[]\T1/pcr/m/n/10.95 list2exp($\OML/cmm/m/it/10.95 L; x$\T1/pcr/m/n/10.95 ) \T1/
ptm/m/n/10.95 re-turns the boolean ex-pres-sion \T1/pcr/m/n/10.95 (($\OML/cmm/m
/it/10.95 x \U/stmry/m/n/10.95 = \OML/cmm/m/it/10.95 a[]\U/stmry/m/n/10.95 )$ \
T1/pcr/m/n/10.95 or ...($\OML/cmm/m/it/10.95 x \U/stmry/m/n/10.95 =
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[87]
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\T1/ptm/m/n/10.95 To check the an-swer, in-put \T1/pcr/m/n/10.95 0173 \T1/ptm/m
/n/10.95 (see Sec-tion [][]6.4.1[][] p.[][]86[][]) or \T1/pcr/m/n/10.95 horner(
revlist([3,7,1]),8)
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\T1/ptm/m/n/10.95 (see Sec-tion [][]6.27.19[][] p.[][]311[][] and Sec-tion [][]
6.40.15[][] p.[][]419[][]) or \T1/pcr/m/n/10.95 convert([3,7,1],base,8)\T1/ptm/
m/n/10.95 .
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[88]
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[]   \T1/pcr/m/n/10.95 697920827223758251185210916864000000000000000000000000[]
 
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[89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99]
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\T1/ptm/m/n/10.95 go-rithm and Miller-Rabin's Al-go-rithm in the Al-go-rith-mic
 part (menu \T1/pcr/m/n/10.95 Help$\U/msa/m/n/10.95 I$\T1/pcr/m/n/10.95 Manuals
$\U/msa/m/n/10.95 I$\T1/pcr/m/n/10.95 Programming\T1/ptm/m/n/10.95 )).
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[102]

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[]\T1/ptm/b/n/12 The num-ber of pseudo-primes less than or equal to $\OML/cmm/m
/it/12 n$\T1/ptm/b/n/12 : \T1/pcr/b/n/12 nprimes
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[103]

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[104]
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[]\T1/pcr/m/n/10.95 ichinrem($\U/stmry/m/n/10.95 [\OML/cmm/m/it/10.95 a[]; p[]\
U/stmry/m/n/10.95 ]$\T1/pcr/m/n/10.95 ,$\U/stmry/m/n/10.95 [\OML/cmm/m/it/10.95
 a[]; p[]\U/stmry/m/n/10.95 ]$\T1/pcr/m/n/10.95 ,...,$\U/stmry/m/n/10.95 [\OML/
cmm/m/it/10.95 a[]; p[]\U/stmry/m/n/10.95 ]$\T1/pcr/m/n/10.95 ) \T1/ptm/m/n/10.
95 if pos-si-ble re-turns a list \T1/pcr/m/n/10.95 [$\OML/cmm/m/it/10.95 c$\T1/
pcr/m/n/10.95 ,$\OML/cmm/m/it/10.95 L$\T1/pcr/m/n/10.95 ]\T1/ptm/m/n/10.95 ,
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\T1/pcr/m/n/10.95 chrem([a,b],[p,q])=ichrem([a,p],[b,q])=ichinrem([a,p],[b,q]) 

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[107]
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\T1/ptm/m/n/10.95 In other words the set of in-te-gers less than 21 and co-prim
e with 21, $\OMS/cmsy/m/n/10.95 f\U/stmry/m/n/10.95 2\OML/cmm/m/it/10.95 ; \U/s
tmry/m/n/10.95 4\OML/cmm/m/it/10.95 ; \U/stmry/m/n/10.95 5\OML/cmm/m/it/10.95 ;
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.95 ; \U/stmry/m/n/10.95 10\OML/cmm/m/it/10.95 ; \U/stmry/m/n/10.95 11\OML/cmm/
m/it/10.95 ; \U/stmry/m/n/10.95 13\OML/cmm/m/it/10.95 ; \U/stmry/m/n/10.95 15\O
ML/cmm/m/it/10.95 ; \U/stmry/m/n/10.95 16\OML/cmm/m/it/10.95 ; \U/stmry/m/n/10.
95 17\OML/cmm/m/it/10.95 ; \U/stmry/m/n/10.95 19\OMS/cmsy/m/n/10.95 g$\T1/ptm/m
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\T1/ptm/m/n/10.95 with \T1/ptm/m/it/10.95 bool \T1/ptm/m/n/10.95 equal to \T1/p
cr/m/n/10.95 true\T1/ptm/m/n/10.95 . With \T1/ptm/m/it/10.95 bool \T1/ptm/m/n/1
0.95 equal to \T1/pcr/m/n/10.95 false\T1/ptm/m/n/10.95 , \T1/pcr/m/n/10.95 icom
p($\OML/cmm/m/it/10.95 n$\T1/pcr/m/n/10.95 ,$\OML/cmm/m/it/10.95 k$\T1/pcr/m/n/
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 \T1/pcr/m/n/10.95 wz_certificate((-1)^^Bk*comb(n,k)*comb(2k,k)*4^^B(n-k),comb(
2n,n),n,k) 
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[]\T1/ptm/b/n/12 Transform a float-ing point num-ber into a ra-tio-nal: \T1/pcr
/b/n/12 exact float2rational
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[]\T1/ptm/b/n/12 Denominator of a frac-tion af-ter sim-pli-fi-ca-tion: \T1/pcr/
b/n/12 denom getDenom[]
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\T1/ptm/m/n/10.95 which ap-prox-i-mates \T1/pcr/m/n/10.95 a \T1/ptm/m/n/10.95 o
r \T1/pcr/m/n/10.95 evalf(a) \T1/ptm/m/n/10.95 with the spec-i-fied pre-ci-sion
 \T1/pcr/m/n/10.95 epsilon\T1/ptm/m/n/10.95 . 
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\T1/ptm/m/n/10.95 equa-tion of or-der 2 with in-te-ger co-ef-fi-cients. So If \
T1/pcr/m/n/10.95 dfc(a)=[a0,a1,a2,[b0,b1]]
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[]\T1/ptm/b/n/12 Transform a con-tin-ued frac-tion rep-re-sen-ta-tion into a re
al: \T1/pcr/b/n/12 dfc2f
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\T1/ptm/m/n/10.95 Indeed \T1/pcr/m/n/10.95 Gamma(0.7)=-0.3*Gamma(-0.3)=(-0.3)*(
-1.3)*Gamma(-1.3) 
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[]\T1/pcr/m/n/10.95 nextperm($\OML/cmm/m/it/10.95 p$\T1/pcr/m/n/10.95 ) \T1/ptm
/m/n/10.95 re-turns the next per-mu-ta-tion in lex-i-co-graphic or-der, or \T1/
pcr/m/n/10.95 undef
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[]\T1/ptm/b/n/12 Multiplication by the com-plex con-ju-gate: \T1/pcr/b/n/12 mul
t_c_conjugate
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[]\T1/pcr/m/n/10.95 barycenter($\OML/cmm/m/it/10.95 l[]; l[]; [] ; l[]$\T1/pcr/
m/n/10.95 ) \T1/ptm/m/n/10.95 re-turns the barycen-ter of the points $\OML/cmm/
m/it/10.95 A[]$ \T1/ptm/m/n/10.95 weighted
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\T1/ptm/m/n/10.95 If $\OML/cmm/m/it/10.95 a$ \T1/ptm/m/n/10.95 is a vari-able, 
then $\OML/cmm/m/it/10.95 a$\T1/pcr/m/n/10.95 :=quote($\OML/cmm/m/it/10.95 a$\T
1/pcr/m/n/10.95 ) \T1/ptm/m/n/10.95 (or $\OML/cmm/m/it/10.95 a$\T1/pcr/m/n/10.9
5 :=hold($\OML/cmm/m/it/10.95 a$\T1/pcr/m/n/10.95 )\T1/ptm/m/n/10.95 ) is equiv
-a-lent to \T1/pcr/m/n/10.95 purge($\OML/cmm/m/it/10.95 a$\T1/pcr/m/n/10.95 )
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\T1/ptm/m/n/10.95 To fac-tor over $\U/msb/m/n/10.95 C$\T1/ptm/m/n/10.95 , put \
T1/pcr/m/n/10.95 Xcas \T1/ptm/m/n/10.95 back in com-plex mode or in-put \T1/pcr
/m/n/10.95 cfactor(x^^B4+1,sqrt(2))\T1/ptm/m/n/10.95 . 
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[]\T1/pcr/m/n/10.95 normal(\T1/ptm/m/it/10.95 expr\T1/pcr/m/n/10.95 ) \T1/ptm/m
/n/10.95 re-turns the ex-panded ir-re-ducible rep-re-sen-ta-tion of \T1/ptm/m/i
t/10.95 expr\T1/ptm/m/n/10.95 . (See
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\T1/ptm/m/n/10.95 also \T1/pcr/m/n/10.95 ratnormal\T1/ptm/m/n/10.95 , Sec-tion 
[][]6.12.16[][] p.[][]167[][], for pure ra-tio-nal func-tion or \T1/pcr/m/n/10.
95 simplify\T1/ptm/m/n/10.95 ,
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[]\T1/ptm/m/it/10.95 Input (if the vari-ables \T1/pcr/m/sl/10.95 a \T1/ptm/m/it
/10.95 and \T1/pcr/m/sl/10.95 b \T1/ptm/m/it/10.95 are purged, oth-er-wise firs
t en-ter \T1/pcr/m/sl/10.95 purge(a,b)\T1/ptm/m/it/10.95 ): 
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[]\T1/ptm/m/it/10.95 Input (if the vari-ables \T1/pcr/m/sl/10.95 a \T1/ptm/m/it
/10.95 and \T1/pcr/m/sl/10.95 b \T1/ptm/m/it/10.95 are purged, oth-er-wise firs
t en-ter \T1/pcr/m/sl/10.95 purge(a,b)\T1/ptm/m/it/10.95 ): 
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[]\T1/ptm/m/n/10.95 In \T1/pcr/m/n/10.95 Mupad \T1/ptm/m/n/10.95 or \T1/pcr/m/n
/10.95 Xcas \T1/ptm/m/n/10.95 or \T1/pcr/m/n/10.95 TI \T1/ptm/m/n/10.95 modes, 
\T1/pcr/m/n/10.95 subs \T1/ptm/m/n/10.95 be-haves like \T1/pcr/m/n/10.95 subst 
\T1/ptm/m/n/10.95 (see Sec-tion [][]6.12.18[][]
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[]\T1/ptm/b/n/12 Eliminate one or more vari-ables from a list of equa-tions: \T
1/pcr/b/n/12 eliminate
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[]\T1/pcr/m/n/10.95 eliminate(\T1/ptm/m/it/10.95 eqns\T1/pcr/m/n/10.95 ,\T1/ptm
/m/it/10.95 vars\T1/pcr/m/n/10.95 ) \T1/ptm/m/n/10.95 re-turns the equa-tions w
ith the vari-ables \T1/ptm/m/it/10.95 vars \T1/ptm/m/n/10.95 elim-
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\T1/ptm/m/n/10.95 grand $\OML/cmm/m/it/10.95 f$ \T1/ptm/m/n/10.95 is known. As-
sume, for ex-am-ple, that \T1/pcr/m/n/10.95 F:=int(f,x)\T1/ptm/m/n/10.95 , then
 \T1/pcr/m/n/10.95 preval(F,a,b)
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[]\T1/pcr/m/n/10.95 tableseq($\OML/cmm/m/it/10.95 f\U/stmry/m/n/10.95 (\OML/cmm
/m/it/10.95 x[]; [] ; x[]\U/stmry/m/n/10.95 )$\T1/pcr/m/n/10.95 ,$\U/stmry/m/n/
10.95 [\OML/cmm/m/it/10.95 x[]; [] ; x[]\U/stmry/m/n/10.95 ]$\T1/pcr/m/n/10.95 
,$\U/stmry/m/n/10.95 [\OML/cmm/m/it/10.95 u[]; [] ; u[]\U/stmry/m/n/10.95 ]$\T1
/pcr/m/n/10.95 ) \T1/ptm/m/n/10.95 fills the cur-rent col-
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[]\T1/ptm/m/n/10.95 The \T1/pcr/m/n/10.95 tan[] \T1/ptm/m/n/10.95 com-mand is t
he tan-gent func-tion (\T1/pcr/m/n/10.95 tan(x)= sin(x)/cos(x)\T1/ptm/m/n/10.95
 ). 
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[]\T1/ptm/m/n/10.95 The \T1/pcr/m/n/10.95 cot[] \T1/ptm/m/n/10.95 com-mand is t
he cotan-gent func-tion (\T1/pcr/m/n/10.95 cot(x)= cos(x)/sin(x)\T1/ptm/m/n/10.
95 ). 
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[]\T1/ptm/m/n/10.95 The \T1/pcr/m/n/10.95 csc[] \T1/ptm/m/n/10.95 com-mand is t
he cose-cant func-tion (\T1/pcr/m/n/10.95 csc(x) = 1/sin(x)\T1/ptm/m/n/10.95 ).
 
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\T1/ptm/m/n/10.95 It is also pos-si-ble to put \T1/pcr/m/n/10.95 x=a \T1/ptm/m/
n/10.95 as ar-gu-ment in-stead of \T1/pcr/m/n/10.95 x,a\T1/ptm/m/n/10.95 ; \T1/
pcr/m/n/10.95 limit(\T1/ptm/m/it/10.95 expr,var\T1/pcr/m/n/10.95 =\T1/ptm/m/it/
10.95 pt[,side]\T1/pcr/m/n/10.95 )
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 \T1/pcr/m/n/10.95 limit((sqrt(1+x+x^^B2/2)-exp(x/2))/((1-cos(x))*sin(x)),x,0) 

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\T1/ptm/m/n/10.95 sion deriva-tive \T1/pcr/m/n/10.95 diff \T1/ptm/m/n/10.95 (se
e Sec-tion [][]6.19.4[][] p.[][]223[][]) in con-junc-tion with \T1/pcr/m/n/10.9
5 unapply
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\T1/ptm/m/n/10.95 length of the curve de-fined by $\OML/cmm/m/it/10.95 y \U/stm
ry/m/n/10.95 = \OML/cmm/m/it/10.95 f\U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 x\U
/stmry/m/n/10.95 ) =$\T1/ptm/m/it/10.95 expr \T1/ptm/m/n/10.95 (resp. by $\OML/
cmm/m/it/10.95 x[] \U/stmry/m/n/10.95 =$\T1/ptm/m/it/10.95 expr$[]$\T1/ptm/m/n/
10.95 ,$\OML/cmm/m/it/10.95 x[] \U/stmry/m/n/10.95 =$\T1/ptm/m/it/10.95 expr$[]
$\T1/ptm/m/n/10.95 )
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 \T1/pcr/m/n/10.95 fMax((x-2)^^B2+(y-1)^^B2,[-.25x^^B2-y^^B2+1>=0,x-2y+1=0],[x,
y],[.5,.75]) 
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[]\T1/ptm/m/it/10.95 expr\T1/ptm/m/n/10.95 , a dif-fer-en-tiable ex-pres-sion i
n-volv-ing in-de-pen-dent vari-ables $\OML/cmm/m/it/10.95 x[]; x[]; [] ; x[]$
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[]\T1/ptm/m/it/10.95 expr\T1/ptm/m/n/10.95 , a dif-fer-en-tiable ex-pres-sion i
n-volv-ing in-de-pen-dent vari-ables $\OML/cmm/m/it/10.95 x[]; x[]; [] ; x[]$
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\T1/ptm/m/n/10.95 This re-sult comes from the de-com-po-si-tion of $[]$ (see Se
c-tion [][]6.32.9[][]
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\U/stmry/m/n/10.95 [0\OML/cmm/m/it/10.95 ; \U/stmry/m/n/10.95 1]$\T1/ptm/m/n/10
.95 , is the def-i-nite in-te-gral) or re-turns \T1/pcr/m/n/10.95 "it is probab
ly not a Riemann
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 \T1/pcr/m/n/10.95 limit(integrate(x/(x^^B2-1)+log((x+1)/(x-1)),x,2,a),a,+infin
ity) 
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\U/stmry/m/n/10.95 1\OML/cmm/m/it/10.95 =x \OMS/cmsy/m/n/10.95 ^^@ \OML/cmm/m/i
t/10.95 x=\U/stmry/m/n/10.95 2 \OMS/cmsy/m/n/10.95 ^^T []\U/stmry/m/n/10.95 (\O
ML/cmm/m/it/10.95 x\U/stmry/m/n/10.95 )\OML/cmm/m/it/10.95 =x \OMS/cmsy/m/n/10.
95 ^^T \U/stmry/m/n/10.95 1\OML/cmm/m/it/10.95 =x$\T1/ptm/m/n/10.95 , and so $[
][] \U/stmry/m/n/10.95 1\OML/cmm/m/it/10.95 =x \OMS/cmsy/m/n/10.95 ^^@ \OML/cmm
/m/it/10.95 x=\U/stmry/m/n/10.95 2\OML/cmm/m/it/10.95 dx \OMS/cmsy/m/n/10.95 ^^
T [][] []\U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 x\U/stmry/m/n/10.95 )\OML/cmm/
m/it/10.95 =xdx \OMS/cmsy/m/n/10.95 ^^T
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\T1/ptm/m/n/10.95 Recall that the di-ver-gence of a vec-tor field $\OT1/cmr/bx/
n/10.95 F \U/stmry/m/n/10.95 = [\OML/cmm/m/it/10.95 F[]; [] ; F[]\U/stmry/m/n/1
0.95 ]$ \T1/ptm/m/n/10.95 with vari-ables $\U/stmry/m/n/10.95 [\OML/cmm/m/it/10
.95 x[]; [] ; x[]\U/stmry/m/n/10.95 ]$
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\T1/ptm/m/n/10.95 The curl of a three-dimensional vec-tor field $\OT1/cmr/bx/n/
10.95 F \U/stmry/m/n/10.95 = [\OML/cmm/m/it/10.95 F[]; F[]; F[]\U/stmry/m/n/10.
95 ]$ \T1/ptm/m/n/10.95 with vari-ables $\U/stmry/m/n/10.95 [\OML/cmm/m/it/10.9
5 x[]; x[]; x[]\U/stmry/m/n/10.95 ]$
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[]\T1/ptm/m/n/10.95 The \T1/pcr/m/n/10.95 vpotential \T1/ptm/m/n/10.95 com-mand
 finds a par-tic-u-lar vec-tor field $\OT1/cmr/bx/n/10.95 G$ \T1/ptm/m/n/10.95 
such that $[]\OT1/cmr/bx/n/10.95 G \U/stmry/m/n/10.95 =
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\T1/ptm/m/n/10.95 only ra-tio-nal nor-mal-iza-tion is per-formed (us-ing the \T
1/pcr/m/n/10.95 ratnormal \T1/ptm/m/n/10.95 com-mand). \T1/pcr/m/n/10.95 simpli
fy=true
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 \T1/pcr/m/n/10.95 convex(x1^^B3+2x1^^B2+2*x1*x2+x2^^B2/2-8x1-2x2-8,[x1,x2]) 
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\T1/ptm/m/n/10.95 plac-ing both op-tional ar-gu-ments by ei-ther, for ex-am-ple
, $\OML/cmm/m/it/10.95 y\U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 x\U/stmry/m/n/1
0.95 )$ \T1/ptm/m/n/10.95 or $\U/stmry/m/n/10.95 [\OML/cmm/m/it/10.95 y[]\U/stm
ry/m/n/10.95 (\OML/cmm/m/it/10.95 x\U/stmry/m/n/10.95 )\OML/cmm/m/it/10.95 ; y[
]\U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 x\U/stmry/m/n/10.95 )\OML/cmm/m/it/10.
95 ; [] ; y[]\U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 x\U/stmry/m/n/10.95 )]$\T1
/ptm/m/n/10.95 . 
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\T1/ptm/m/n/10.95 func-tion for the prob-lem of min-i-miz-ing some func-tional 
$\OML/cmm/m/it/10.95 F\U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 y\U/stmry/m/n/10.
95 ) = [][] \OML/cmm/m/it/10.95 f\U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 x; y; 
y[]\U/stmry/m/n/10.95 ) d\OML/cmm/m/it/10.95 x$\T1/ptm/m/n/10.95 . 
 []

[254]
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\T1/ptm/m/n/10.95 First, set $\OML/cmm/m/it/10.95 f\U/stmry/m/n/10.95 (\OML/cmm
/m/it/10.95 y; y[]; x\U/stmry/m/n/10.95 ) = \OML/cmm/m/it/10.95 y\U/stmry/m/n/1
0.95 (\OML/cmm/m/it/10.95 x\U/stmry/m/n/10.95 ) []$ \T1/ptm/m/n/10.95 and com-p
ute the as-so-ci-ated Euler-Lagrange
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[255]
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\T1/ptm/m/n/10.95 The func-tion $\OML/cmm/m/it/10.95 y[]$ \T1/ptm/m/n/10.95 de-
pends on two pa-ram-e-ters, so you can use \T1/pcr/m/n/10.95 conjugate_equation

 []

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[]\T1/ptm/b/n/12 Increase the phase by $\OML/cmm/m/it/12 ^^Y=\U/stmry/m/n/12 2$
 \T1/ptm/b/n/12 in a trigono-met-ric ex-pres-sion: \T1/pcr/b/n/12 shift_phase
 []


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[259] [260]
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\T1/ptm/m/n/10.95 ready, by check-ing \T1/pcr/m/n/10.95 radian \T1/ptm/m/n/10.9
5 in the \T1/pcr/m/n/10.95 cas \T1/ptm/m/n/10.95 con-fig-u-ra-tion or in-puttin
g \T1/pcr/m/n/10.95 angle_radian:=1\T1/ptm/m/n/10.95 ,
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\T1/pcr/m/n/10.95 tlin \T1/ptm/m/n/10.95 (see Sec-tion [][]6.23.2[][] p.[][]259
[][]), \T1/pcr/m/n/10.95 tcollect \T1/ptm/m/n/10.95 (see Sec-tion [][]6.23.4[][
] p.[][]261[][]), \T1/pcr/m/n/10.95 trigsin
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[261]
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 \T1/pcr/m/n/10.95 trigsimplify((sin(x+y)-sin(x-y))/(cos(x+y)+cos(x-y))) 
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[262] [263] [264]
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 \T1/pcr/m/n/10.95 simplify(sincos(((i)*(exp((i)*x))^^B2-i)/(2*exp((i)*x)))) 
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 \T1/pcr/m/n/10.95 simplify(exp2trig(((i)*(exp((i)*x))^^B2-i)/(2*exp((i)*x)))) 

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[265] [266] [267]
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\T1/ptm/b/n/12 hy-per-bolic func-tions as func-tion of exp(x): \T1/pcr/b/n/12 h
alftan_hyp2exp
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[268]
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 \T1/pcr/m/n/10.95 halftan_hyp2exp(sin(x)^^B2+cos(x)^^B2-sinh(x)^^B2+cosh(x)^^B
2) 
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[269] [270]
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\T1/ptm/m/n/10.95 The \T1/pcr/m/n/10.95 trigtan \T1/ptm/m/n/10.95 com-mand rewr
ites trigono-met-ric ex-pres-sions into ex-pres-sions where
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[]\T1/ptm/b/n/12 Rewrite an ex-pres-sion with dif-fer-ent op-tions: \T1/pcr/b/n
/12 convert convertir
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\T1/pcr/m/n/10.95 Xcas \T1/ptm/m/n/10.95 has many com-mands to con-vert ex-pres
-sions into dif-fer-ent forms; the \T1/pcr/m/n/10.95 convert
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Overfull \hbox (9.77956pt too wide) in paragraph at lines 20703--20706
[]\T1/pcr/m/n/10.95 sincos\T1/ptm/m/n/10.95 , to con-vert an ex-pres-sion like 
\T1/pcr/m/n/10.95 sincos \T1/ptm/m/n/10.95 (see Sec-tion [][]6.23.13[][]
 []


Overfull \hbox (3.20956pt too wide) in paragraph at lines 20712--20715
[]\T1/pcr/m/n/10.95 exp\T1/ptm/m/n/10.95 , to con-vert an ex-pres-sion like \T1
/pcr/m/n/10.95 trig2exp \T1/ptm/m/n/10.95 (see Sec-tion [][]6.23.21[][]
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[271]
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[]\T1/pcr/m/n/10.95 expln\T1/ptm/m/n/10.95 , to con-vert an ex-pres-sion like \
T1/pcr/m/n/10.95 trig2exp \T1/ptm/m/n/10.95 (see Sec-tion [][]6.23.21[][]
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[272]
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\T1/ptm/m/n/10.95 the \T1/pcr/m/n/10.95 hyp2exp \T1/ptm/m/n/10.95 com-mand con-
verts hy-per-bolic func-tions into their ex-po-nen-tial forms. 
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[273] [274] [275]
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[]\T1/pcr/m/n/10.95 pow2exp(\T1/ptm/m/it/10.95 expr\T1/pcr/m/n/10.95 ) \T1/ptm/
m/n/10.95 re-turns \T1/ptm/m/it/10.95 expr \T1/ptm/m/n/10.95 with any pow-ers r
e-placed by their cor-re-spond-
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[276]
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[]\T1/ptm/b/n/12 Expand a tran-scen-den-tal and trigono-met-ric ex-pres-sion: \
T1/pcr/b/n/12 texpand
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\T1/ptm/m/sl/10.95 6.25.  REWRITING TRANSCENDENTAL AND TRIGONOMETRIC EXPRESSION
S \T1/ptm/m/n/10.95 277
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[]\T1/pcr/m/n/10.95 combine(\T1/ptm/m/it/10.95 expr\T1/pcr/m/n/10.95 ,trig) \T1
/ptm/m/n/10.95 or \T1/pcr/m/n/10.95 combine(\T1/ptm/m/it/10.95 expr\T1/pcr/m/n/
10.95 ,sin) \T1/ptm/m/n/10.95 or \T1/pcr/m/n/10.95 combine(\T1/ptm/m/it/10.95 e
xpr,cos)
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\T1/ptm/m/sl/10.95 6.25.  REWRITING TRANSCENDENTAL AND TRIGONOMETRIC EXPRESSION
S \T1/ptm/m/n/10.95 279
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[]\T1/ptm/b/n/12 Fourier co-ef-fi-cients: \T1/pcr/b/n/12 fourier_an \T1/ptm/b/n
/12 and \T1/pcr/b/n/12 fourier_bn \T1/ptm/b/n/12 or \T1/pcr/b/n/12 fourier_cn
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[280] [281] [282] [283]
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[]\T1/ptm/b/n/12 Continuous Fourier Trans-form: \T1/pcr/b/n/12 fourier ifourier
 addtable
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[]\T1/pcr/m/n/10.95 fourier(\T1/ptm/m/it/10.95 expr$ \OMS/cmsy/m/n/10.95 h\OML/
cmm/m/it/10.95 ; x; s\OMS/cmsy/m/n/10.95 i$\T1/ptm/m/n/10.95 ) re-turns the Fou
rier trans-form $\OML/cmm/m/it/10.95 F\U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 s
\U/stmry/m/n/10.95 )$\T1/ptm/m/n/10.95 . If $\OML/cmm/m/it/10.95 s$ \T1/ptm/m/n
/10.95 is not given,
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[284]
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Overfull \hbox (7.1422pt too wide) in paragraph at lines 22016--22023
\T1/ptm/m/n/10.95 You can ob-tain the orig-i-nal func-tion $\OML/cmm/m/it/10.95
 f$ \T1/ptm/m/n/10.95 from the above re-sult by ap-ply-ing \T1/pcr/m/n/10.95 if
ourier\T1/ptm/m/n/10.95 . 
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Overfull \hbox (11.62187pt too wide) in paragraph at lines 22038--22046
\T1/ptm/m/n/10.95 Fourier trans-form. For ex-am-ple, if $\OML/cmm/m/it/10.95 g\
U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 x\U/stmry/m/n/10.95 ) = \OML/cmm/m/it/10
.95 f\U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 x \OMS/cmsy/m/n/10.95 ^^@ \OML/cmm
/m/it/10.95 a\U/stmry/m/n/10.95 )$\T1/ptm/m/n/10.95 , then $\OMS/cmsy/m/n/10.95
 F\U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 g\U/stmry/m/n/10.95 )(\OML/cmm/m/it/1
0.95 s\U/stmry/m/n/10.95 ) = \OML/cmm/m/it/10.95 e[]\OMS/cmsy/m/n/10.95 F\U/stm
ry/m/n/10.95 (\OML/cmm/m/it/10.95 f\U/stmry/m/n/10.95 )(\OML/cmm/m/it/10.95 s\U
/stmry/m/n/10.95 )$\T1/ptm/m/n/10.95 .
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[290]
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\T1/ptm/m/n/10.95 If $\OML/cmm/m/it/10.95 x$ \T1/ptm/m/n/10.95 is a pe-ri-odic 
se-quence of pe-riod $\OML/cmm/m/it/10.95 N$\T1/ptm/m/n/10.95 , de-fined by the
 vec-tor $\OML/cmm/m/it/10.95 x \U/stmry/m/n/10.95 = [\OML/cmm/m/it/10.95 x[]; 
x[]; [] x[]\U/stmry/m/n/10.95 ]$
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\T1/ptm/m/n/10.95 Again, the co-ef-fi-cients of $\OML/cmm/m/it/10.95 P$ \T1/ptm
/m/n/10.95 are [0,1,1,0], $\OML/cmm/m/it/10.95 N \U/stmry/m/n/10.95 = 4$ \T1/pt
m/m/n/10.95 and $\OML/cmm/m/it/10.95 ! \U/stmry/m/n/10.95 = [](2\OML/cmm/m/it/1
0.95 i^^Y=\U/stmry/m/n/10.95 4) =
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\T1/ptm/m/n/10.95 Therefore, if $\OML/cmm/m/it/10.95 m \OMS/cmsy/m/n/10.95 ^^@ 
\OML/cmm/m/it/10.95 n$ \T1/ptm/m/n/10.95 is a mul-ti-ple of $\OML/cmm/m/it/10.9
5 N$ \T1/ptm/m/n/10.95 ($\OML/cmm/m/it/10.95 m \U/stmry/m/n/10.95 = \OML/cmm/m/
it/10.95 n \U/stmry/m/n/10.95 + \OML/cmm/m/it/10.95 l \OMS/cmsy/m/n/10.95 ^^A \
OML/cmm/m/it/10.95 N$\T1/ptm/m/n/10.95 ) then $[][] \OML/cmm/m/it/10.95 ![] \U/
stmry/m/n/10.95 =
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\OML/cmm/m/it/10.95 q \U/stmry/m/n/10.95 = [\OML/cmm/m/it/10.95 q[]; [] q[]\U/s
tmry/m/n/10.95 ] = [\OML/cmm/m/it/10.95 p[]; ::p[]; p[]; ::; p[]\U/stmry/m/n/10
.95 ] = []\OML/cmm/m/it/10.95 F[]\U/stmry/m/n/10.95 ([\OML/cmm/m/it/10.95 y[]; 
::y[]\U/stmry/m/n/10.95 ]) = []
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[298]
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\T1/ptm/m/n/10.95 the vari-able list \T1/pcr/m/n/10.95 [x,y]\T1/ptm/m/n/10.95 )
, and $\U/stmry/m/n/10.95 2\OML/cmm/m/it/10.95 x[]y[]z \OMS/cmsy/m/n/10.95 ^^@ 
\U/stmry/m/n/10.95 5\OML/cmm/m/it/10.95 xz$ \T1/ptm/m/n/10.95 is rep-re-sented 
by \T1/pcr/m/n/10.95 %%%{2,[3,2,1]%%%}
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[299]
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\T1/ptm/m/n/10.95 mial which re-places each term \T1/pcr/m/n/10.95 %%%{$\OML/cm
m/m/it/10.95 a$\T1/pcr/m/n/10.95 ,[$\OML/cmm/m/it/10.95 n[]; [] ; n[]$\T1/pcr/m
/n/10.95 ] %%%} \T1/ptm/m/n/10.95 in $\OML/cmm/m/it/10.95 P$ \T1/ptm/m/n/10.95 
by \T1/pcr/m/n/10.95 %%%{$\OML/cmm/m/it/10.95 f\U/stmry/m/n/10.95 (\OML/cmm/m/i
t/10.95 a; n[]; [] ; n[]\U/stmry/m/n/10.95 )$\T1/pcr/m/n/10.95 ,[$\OML/cmm/m/it
/10.95 n[]; [] ; n[]$\T1/pcr/m/n/10.95 ]
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[]$\OML/cmm/m/it/10.95 S$\T1/ptm/m/n/10.95 , a sum of mono-mi-als of the form \
T1/pcr/m/n/10.95 %%%{coeff,[n1,...nk] %%%} 
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[300]
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 \T1/pcr/m/n/10.95 r2e(%%%{1,[2,0]%%%}+%%%{-1,[1,1]%%%}+%%%{2,[0,1]%%%},[x,y]) 

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Overfull \hbox (94.42455pt too wide) in paragraph at lines 22874--22875
 \T1/pcr/m/n/10.95 poly2symb(%%%{1,[2,0]%%%}+%%%{-1,[1,1]%%%}+%%%{2,[0,1]%%%},[
x,y]) 
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[301]
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\T1/ptm/m/n/10.95 The \T1/pcr/m/n/10.95 convert \T1/ptm/m/n/10.95 com-mand does
 many con-ver-sions (see Sec-tion [][]6.23.26[][] p.[][]271[][]). Among
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[303] [304] [305] [306]

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[307] [308] [309] [310] [311] [312]
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$\OML/cmm/m/it/10.95 P\U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 x\U/stmry/m/n/10.
95 ) = \OML/cmm/m/it/10.95 x$ \T1/ptm/m/n/10.95 is rep-re-sented by [1,0] and $
\OML/cmm/m/it/10.95 $ \T1/ptm/m/n/10.95 by \T1/pcr/m/n/10.95 rootof([1,0],[1,0
,10,0,1])\T1/ptm/m/n/10.95 . 
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[]\T1/ptm/b/n/12 Convert a se-ries ex-pan-sion into a poly-no-mial: \T1/pcr/b/n
/12 convert convertir[]
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[]\T1/pcr/m/n/10.95 reorder(\T1/ptm/m/it/10.95 expr,vars\T1/pcr/m/n/10.95 ) \T1
/ptm/m/n/10.95 ex-pands \T1/ptm/m/it/10.95 expr \T1/ptm/m/n/10.95 ac-cord-ing t
o the or-der of vari-ables given
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[317]
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\T1/ptm/m/n/10.95 sec-ond mem-ber of the as-sign-ment is not eval-u-ated). Note
 also that \T1/pcr/m/n/10.95 g(x):=lagrange([1,2],[3,4])
 []

[318]
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[]\T1/pcr/m/n/10.95 triginterp($\OML/cmm/m/it/10.95 L; a; b; x$\T1/pcr/m/n/10.9
5 ) \T1/ptm/m/n/10.95 or \T1/pcr/m/n/10.95 triginterp($\OML/cmm/m/it/10.95 L; x
 \U/stmry/m/n/10.95 = \OML/cmm/m/it/10.95 a::b$\T1/pcr/m/n/10.95 ) \T1/ptm/m/n/
10.95 re-turns the trigono-
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\T1/ptm/m/n/10.95 kinds of in-ter-po-la-tion ob-tained by spec-i-fy-ing three k
inds of con-straints. The unique-
 []

[321]
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[]\T1/pcr/m/n/10.95 spline($\OML/cmm/m/it/10.95 L[]; L[]; x; l$\T1/pcr/m/n/10.9
5 ) \T1/ptm/m/n/10.95 re-turns the nat-u-ral spline func-tion $\OML/cmm/m/it/10
.95 s$ \T1/ptm/m/n/10.95 of de-gree $\OML/cmm/m/it/10.95 l$\T1/ptm/m/n/10.95 , 
where
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[322]
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\U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 y[]; y[]; [] ; y[]\U/stmry/m/n/10.95 )$
 \T1/ptm/m/n/10.95 This method com-putes Thiele in-ter-po-lated con-tin-ued fun
c-tion based
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[323]
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 [] 
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[]
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[324] [325] [326] [327]
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[]\T1/pcr/m/n/10.95 gcd(\T1/ptm/m/it/10.95 polys\T1/pcr/m/n/10.95 ) \T1/ptm/m/n
/10.95 re-turns the great-est com-mon di-vi-sor of the poly-no-mi-als in \T1/pt
m/m/it/10.95 polys\T1/ptm/m/n/10.95 . 
 []

[328]
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[]\T1/ptm/b/n/12 Choosing the GCD al-go-rithm of two poly-no-mi-als: \T1/pcr/b/
n/12 ezgcd heugcd
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[329] [330] [331] [332] [333]
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[336]
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\T1/ptm/m/n/10.95 nom-i-na-tor (here 1) and the Sturm se-quence of the de-nom-i
-na-tor ([[3,0,2],[6,0],-
 []

[337] [338]
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[]\T1/ptm/b/n/12 Sylvester ma-trix of two poly-no-mi-als and re-sul-tant: \T1/p
cr/b/n/12 sylvester
 []


Overfull \hbox (2.54227pt too wide) in paragraph at lines 25869--25874
\T1/ptm/m/n/10.95 Given two poly-no-mi-als $\OML/cmm/m/it/10.95 A\U/stmry/m/n/1
0.95 (\OML/cmm/m/it/10.95 x\U/stmry/m/n/10.95 ) = [][] \OML/cmm/m/it/10.95 a[]x
[]$ \T1/ptm/m/n/10.95 and $\OML/cmm/m/it/10.95 B\U/stmry/m/n/10.95 (\OML/cmm/m/
it/10.95 x\U/stmry/m/n/10.95 ) = [][] \OML/cmm/m/it/10.95 b[]x[]$\T1/ptm/m/n/10
.95 , their Sylvester
 []


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\T1/ptm/m/n/10.95 ma-trix is a square ma-trix of size $\OML/cmm/m/it/10.95 m \U
/stmry/m/n/10.95 + \OML/cmm/m/it/10.95 n$ \T1/ptm/m/n/10.95 where $\OML/cmm/m/i
t/10.95 m$\T1/ptm/m/n/10.95 =degree($\OML/cmm/m/it/10.95 B\U/stmry/m/n/10.95 (\
OML/cmm/m/it/10.95 x\U/stmry/m/n/10.95 )$\T1/ptm/m/n/10.95 ) and $\OML/cmm/m/it
/10.95 n$\T1/ptm/m/n/10.95 =degree($\OML/cmm/m/it/10.95 A\U/stmry/m/n/10.95 (\O
ML/cmm/m/it/10.95 x\U/stmry/m/n/10.95 )$\T1/ptm/m/n/10.95 ).
 []


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[]
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[]
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\T1/ptm/m/n/10.95 has a unique so-lu-tion $\OML/cmm/m/it/10.95 U; V$ \T1/ptm/m/
n/10.95 such that degree$\U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 U\U/stmry/m/n/
10.95 ) \OML/cmm/m/it/10.95 <$\T1/ptm/m/n/10.95 degree$\U/stmry/m/n/10.95 (\OML
/cmm/m/it/10.95 B\U/stmry/m/n/10.95 )$ \T1/ptm/m/n/10.95 and degree$\U/stmry/m/
n/10.95 (\OML/cmm/m/it/10.95 V\U/stmry/m/n/10.95 ) \OML/cmm/m/it/10.95 <$\T1/pt
m/m/n/10.95 degree$\U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 A\U/stmry/m/n/10.95 
)$\T1/ptm/m/n/10.95 ,
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[340]
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\T1/ptm/m/n/10.95 The re-sul-tant of both equa-tions with re-spect to $\OML/cmm
/m/it/10.95 t$\T1/ptm/m/n/10.95 , \T1/pcr/m/n/10.95 resultant(eq1,eq2,t)\T1/ptm
/m/n/10.95 ,
 []

[341]
Overfull \hbox (28.00572pt too wide) detected at line 26022
\OMS/cmsy/m/n/10.95 ^^@\U/stmry/m/n/10.95 (64 \OMS/cmsy/m/n/10.95 ^^A \U/stmry/
m/n/10.95 (\OML/cmm/m/it/10.95 x[] \U/stmry/m/n/10.95 + \OML/cmm/m/it/10.95 y[]
\U/stmry/m/n/10.95 ) \OMS/cmsy/m/n/10.95 ^^A \U/stmry/m/n/10.95 (\OML/cmm/m/it/
10.95 x[] \OMS/cmsy/m/n/10.95 ^^A \OML/cmm/m/it/10.95 a[] \OMS/cmsy/m/n/10.95 ^
^@ \OML/cmm/m/it/10.95 x[] \OMS/cmsy/m/n/10.95 ^^A \OML/cmm/m/it/10.95 c[] \U/s
tmry/m/n/10.95 + \OMS/cmsy/m/n/10.95 ^^@\U/stmry/m/n/10.95 2 \OMS/cmsy/m/n/10.9
5 ^^A \OML/cmm/m/it/10.95 x \OMS/cmsy/m/n/10.95 ^^A \OML/cmm/m/it/10.95 a[] \OM
S/cmsy/m/n/10.95 ^^A \OML/cmm/m/it/10.95 c \U/stmry/m/n/10.95 + 2 \OMS/cmsy/m/n
/10.95 ^^A \OML/cmm/m/it/10.95 x \OMS/cmsy/m/n/10.95 ^^A \OML/cmm/m/it/10.95 c[
] \OMS/cmsy/m/n/10.95 ^^@ \OML/cmm/m/it/10.95 a[] \U/stmry/m/n/10.95 + 2 \OMS/c
msy/m/n/10.95 ^^A \OML/cmm/m/it/10.95 a[] \OMS/cmsy/m/n/10.95 ^^A \OML/cmm/m/it
/10.95 c[] \U/stmry/m/n/10.95 + \OML/cmm/m/it/10.95 a[] \OMS/cmsy/m/n/10.95 ^^A
 \OML/cmm/m/it/10.95 y[] \OMS/cmsy/m/n/10.95 ^^@ \OML/cmm/m/it/10.95 c[]\U/stmr
y/m/n/10.95 ))
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 [] 
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 \T1/pcr/m/n/10.95 normal(subst(-c^^B2*X^^B2+c^^B2*a^^B2+X^^B2*a^^B2-a^^B4+a^^B
2*Y^^B2,c^^B2=a^^B2-b^^B2)) 
 []

[342] [343]
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[]\T1/ptm/b/n/12 Exact bounds for com-plex roots of a poly-no-mial: \T1/pcr/b/n
/12 complexroot
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[344]
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[]
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Overfull \hbox (59.0718pt too wide) in paragraph at lines 26275--26278
[]$\OML/cmm/m/it/10.95 a$\T1/ptm/m/n/10.95 +i$\OML/cmm/m/it/10.95 b$ \T1/ptm/m/
n/10.95 is a root of mul-ti-plic-ity 1 with $\U/stmry/m/n/10.95 0\OML/cmm/m/it/
10.95 :\U/stmry/m/n/10.95 499999046325680 \OMS/cmsy/m/n/10.95 ^^T \OML/cmm/m/it
/10.95 a \OMS/cmsy/m/n/10.95 ^^T \U/stmry/m/n/10.95 0\OML/cmm/m/it/10.95 :\U/st
mry/m/n/10.95 500000953674320$
 []


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\U/stmry/m/n/10.95 0\OML/cmm/m/it/10.95 :\U/stmry/m/n/10.95 500000953674320$ \T
1/ptm/m/n/10.95 and $\U/stmry/m/n/10.95 0\OML/cmm/m/it/10.95 :\U/stmry/m/n/10.9
5 866024494171135 \OMS/cmsy/m/n/10.95 ^^T \OML/cmm/m/it/10.95 d \OMS/cmsy/m/n/1
0.95 ^^T \U/stmry/m/n/10.95 0\OML/cmm/m/it/10.95 :\U/stmry/m/n/10.95 8660264015
19779$\T1/ptm/m/n/10.95 . 
 []


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[]
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[345]
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[]\T1/ptm/b/n/12 Exact bounds for pos-i-tive real roots of a poly-no-mial: \T1/
pcr/b/n/12 VAS_positive
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[346]
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\T1/ptm/m/n/10.95 The \T1/pcr/m/n/10.95 posubLMQ \T1/ptm/m/n/10.95 com-mand use
s the Lo-cal Max Quadratic (LMQ) Akritas-Strzebonski-
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[347]
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[]\T1/ptm/b/n/12 A lower bound for the pos-i-tive real roots of a poly-no-mial:
 \T1/pcr/b/n/12 poslbdLMQ
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\T1/ptm/m/n/10.95 The \T1/pcr/m/n/10.95 poslbLMQ \T1/ptm/m/n/10.95 com-mand use
s the Lo-cal Max Quadratic (LMQ) Akritas-Strzebonski-
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[]\T1/ptm/b/n/12 Exact val-ues of ra-tio-nal roots of a poly-no-mial: \T1/pcr/b
/n/12 rationalroot
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[348] [349] [350] [351] [352] [353]
Overfull \hbox (1.3003pt too wide) in paragraph at lines 26924--26926
[]\T1/pcr/m/n/10.95 tchebyshev2($\OML/cmm/m/it/10.95 n \OMS/cmsy/m/n/10.95 h\OM
L/cmm/m/it/10.95 ; x\OMS/cmsy/m/n/10.95 i$\T1/pcr/m/n/10.95 ) \T1/ptm/m/n/10.95
 re-turns the Tcheby-chev poly-no-mial of sec-ond kind
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Overfull \hbox (6.37451pt too wide) in paragraph at lines 27021--27027
[]\T1/ptm/m/n/10.95 Optionally, \T1/pcr/m/n/10.95 with_f5=\T1/ptm/m/it/10.95 bo
olean\T1/ptm/m/n/10.95 , where \T1/ptm/m/it/10.95 boolean \T1/ptm/m/n/10.95 can
 be \T1/pcr/m/n/10.95 true \T1/ptm/m/n/10.95 or \T1/pcr/m/n/10.95 false\T1/ptm/
m/n/10.95 .
 []


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[]\T1/pcr/m/n/10.95 gbasis(\T1/ptm/m/it/10.95 polys,vars $ \OMS/cmsy/m/n/10.95 
h$ \T1/ptm/m/it/10.95 ,or-der,\T1/pcr/m/n/10.95 with_cocoa=\T1/ptm/m/it/10.95 b
oolean\T1/ptm/m/n/10.95 , \T1/pcr/m/n/10.95 with_f5=\T1/ptm/m/it/10.95 boolean$
\OMS/cmsy/m/n/10.95 i$\T1/pcr/m/n/10.95 )
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\T1/pcr/m/n/10.95 x^^B2*y^^B3*z^^B4\T1/ptm/m/n/10.95 , but if \T1/ptm/m/it/10.9
5 vars\T1/pcr/m/n/10.95 =[x,z,y]\T1/ptm/m/n/10.95 , then \T1/pcr/m/n/10.95 x^^B
2*y^^B4*z^^B3 \T1/ptm/m/n/10.95 comes af-ter \T1/pcr/m/n/10.95 x^^B2*y^^B3*z^^B
4\T1/ptm/m/n/10.95 . 
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 \T1/pcr/m/n/10.95 greduce(x1^^B2*x3^^B2,[x3^^B3-1,-x2^^B2-x2*x3-x3^^B2,x1+x2+x
3], 
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[]\T1/pcr/m/n/10.95 in_ideal($\OML/cmm/m/it/10.95 P$\T1/ptm/m/it/10.95 ,gbasis,
vars $ \OMS/cmsy/m/n/10.95 h\OML/cmm/m/it/10.95 ;$ \T1/ptm/m/it/10.95 op-tion $
\OMS/cmsy/m/n/10.95 i$\T1/pcr/m/n/10.95 ) \T1/ptm/m/n/10.95 re-turns the value 
\T1/pcr/m/n/10.95 true \T1/ptm/m/n/10.95 (1) or \T1/pcr/m/n/10.95 false
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\T1/ptm/m/n/10.95 The \T1/pcr/m/n/10.95 quo \T1/ptm/m/n/10.95 com-mand finds th
e quo-tient of of two poly-no-mi-als (see also Sec-tion [][]6.28.2[][]
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[]\T1/ptm/m/n/10.95 For poly-no-mi-als, the re-sult of $\OML/cmm/m/it/10.95 P=Q
$ \T1/ptm/m/n/10.95 is its ir-re-ducible rep-re-sen-ta-tion in $\U/msb/m/n/10.9
5 Z\OML/cmm/m/it/10.95 =p\U/msb/m/n/10.95 Z\U/stmry/m/n/10.95 [\OML/cmm/m/it/10
.95 x\U/stmry/m/n/10.95 ]$\T1/ptm/m/n/10.95 . 
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[] \OML/cmm/m/it/10.95 x[] \U/stmry/m/n/10.95 + [] \OML/cmm/m/it/10.95 x[] \U/s
tmry/m/n/10.95 + [] \OML/cmm/m/it/10.95 x[] \U/stmry/m/n/10.95 + [] \OML/cmm/m/
it/10.95 x[] \U/stmry/m/n/10.95 + [] \OML/cmm/m/it/10.95 x \U/stmry/m/n/10.95 +
 1 % 13
 []


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\T1/ptm/m/n/10.95 because $\U/stmry/m/n/10.95 10 = \OMS/cmsy/m/n/10.95 ^^@\U/st
mry/m/n/10.95 3 ( []  13)\OML/cmm/m/it/10.95 ;   \U/stmry/m/n/10.95 40 = 1 ( []
  13)\OML/cmm/m/it/10.95 ;   \U/stmry/m/n/10.95 80 = 2 ( []  13)\OML/cmm/m/it/1
0.95 ;   \U/stmry/m/n/10.95 32 =
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[]\T1/ptm/b/n/12 Rebuild a frac-tion from its value mod-ulo $\OML/cmm/m/it/12 p
$\T1/ptm/b/n/12 : \T1/pcr/b/n/12 fracmod  iratrecon
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[]$\OML/cmm/m/it/10.95 n$\T1/ptm/m/n/10.95 , an in-te-ger greater than 1 (or an
 ir-re-ducible poly-no-mial over $\U/msb/m/n/10.95 Z\OML/cmm/m/it/10.95 =p\U/ms
b/m/n/10.95 Z\U/stmry/m/n/10.95 [\OML/cmm/m/it/10.95 X\U/stmry/m/n/10.95 ]$\T1/
ptm/m/n/10.95 ). 
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[]\T1/ptm/b/n/12 Factorize a poly-no-mial with co-ef-fi-cients in a Ga-lois fie
ld: \T1/pcr/b/n/12 factor
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\T1/ptm/m/n/10.95 clidean quo-tient of two poly-no-mi-als with-out eval-u-a-tio
n. (See sec-tion Sec-tion [][]6.28.2[][]
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\T1/ptm/m/n/10.95 clidean re-main-der of two poly-no-mi-als with-out eval-u-a-t
ion. (See sec-tion Sec-tion [][]6.28.3[][]
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[]\T1/pcr/m/n/10.95 Gcd(\T1/ptm/m/it/10.95 polys\T1/pcr/m/n/10.95 ) \T1/ptm/m/n
/10.95 re-turns the great-est com-mon di-vi-sor of the poly-no-mi-als in \T1/pc
r/m/n/10.95 polys\T1/ptm/m/n/10.95 . 
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[] \OMS/cmsy/m/n/10.95 ^^@ [] [] \U/stmry/m/n/10.95 + 2 [][] + [][] [][] \OMS/c
msy/m/n/10.95 ^^@ [] [][] \U/stmry/m/n/10.95 + [][] [] []
 []

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\OMS/cmsy/m/n/10.95 ^^@\U/stmry/m/n/10.95 2 [][] + 1 + [] + [] [][] \OMS/cmsy/m
/n/10.95 ^^@ [] [][] \U/stmry/m/n/10.95 + [][] [] []
 []

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[]\T1/pcr/m/n/10.95 pade(\T1/ptm/m/it/10.95 expr\T1/pcr/m/n/10.95 ,$\OML/cmm/m/
it/10.95 x; n; p$\T1/pcr/m/n/10.95 ) \T1/ptm/m/n/10.95 or \T1/pcr/m/n/10.95 pad
e(\T1/ptm/m/it/10.95 expr\T1/pcr/m/n/10.95 ,$\OML/cmm/m/it/10.95 x; P; p$\T1/pc
r/m/n/10.95 ) \T1/ptm/m/n/10.95 re-turns a ra-tio-nal func-tion $\OML/cmm/m/it/
10.95 P=Q$
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\T1/ptm/m/n/10.95 If $\OML/cmm/m/it/10.95 a > b$\T1/ptm/m/n/10.95 , then \T1/pc
r/m/n/10.95 i[$\OML/cmm/m/it/10.95 a; b$\T1/pcr/m/n/10.95 ] \T1/ptm/m/n/10.95 r
e-turns \T1/pcr/m/n/10.95 i[evalf($\OML/cmm/m/it/10.95 b$\T1/pcr/m/n/10.95 ,15)
-epsilon,evalf($\OML/cmm/m/it/10.95 a$\T1/pcr/m/n/10.95 ,15)+epsilon]
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\T1/ptm/m/n/10.95 p.[][]79[][], Sec-tion [][]6.15.3[][] p.[][]188[][], Sec-tion
 [][]6.40.4[][] p.[][]411[][], Sec-tion [][]6.38.2[][] p.[][]394[][], Sec-tion 
[][]6.55.4[][]
 []

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[]\T1/pcr/m/n/10.95 remove($\OML/cmm/m/it/10.95 f; L$\T1/pcr/m/n/10.95 ) \T1/pt
m/m/n/10.95 re-turns the sub-list of $\OML/cmm/m/it/10.95 L$ \T1/ptm/m/n/10.95 
with the el-e-ments $\OML/cmm/m/it/10.95 c$ \T1/ptm/m/n/10.95 such that $\OML/c
mm/m/it/10.95 f\U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 c\U/stmry/m/n/10.95 )$\T
1/pcr/m/n/10.95 ==true
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[]\T1/ptm/b/n/12 Concatenate two lists or a list and an el-e-ment: \T1/pcr/b/n/
12 concat augment
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[]\T1/pcr/m/n/10.95 foldr($\OML/cmm/m/it/10.95 R; I; a[]; a[]; [] ; a[]$\T1/pcr
/m/n/10.95 ) \T1/ptm/m/n/10.95 re-turns $\OML/cmm/m/it/10.95 R\U/stmry/m/n/10.9
5 (\OML/cmm/m/it/10.95 a; [] \U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 R\U/stmry/
m/n/10.95 (\OML/cmm/m/it/10.95 a[]; R\U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 a[
]; [] R\U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 a[]; R\U/stmry/m/n/10.95 (\OML/c
mm/m/it/10.95 a[]; I\U/stmry/m/n/10.95 ))))$ 
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\U/stmry/m/n/10.95 [ [[][1\OML/cmm/m/it/10.95 ; \U/stmry/m/n/10.95 3\OML/cmm/m/
it/10.95 ; \U/stmry/m/n/10.95 5]\OML/cmm/m/it/10.95 ; []\U/stmry/m/n/10.95 [1\O
ML/cmm/m/it/10.95 ; \U/stmry/m/n/10.95 3\OML/cmm/m/it/10.95 ; \U/stmry/m/n/10.9
5 6]\OML/cmm/m/it/10.95 ; []\U/stmry/m/n/10.95 [1\OML/cmm/m/it/10.95 ; \U/stmry
/m/n/10.95 4\OML/cmm/m/it/10.95 ; \U/stmry/m/n/10.95 5]\OML/cmm/m/it/10.95 ; []
\U/stmry/m/n/10.95 [1\OML/cmm/m/it/10.95 ; \U/stmry/m/n/10.95 4\OML/cmm/m/it/10
.95 ; \U/stmry/m/n/10.95 6]\OML/cmm/m/it/10.95 ; []\U/stmry/m/n/10.95 [2\OML/cm
m/m/it/10.95 ; \U/stmry/m/n/10.95 3\OML/cmm/m/it/10.95 ; \U/stmry/m/n/10.95 5]\
OML/cmm/m/it/10.95 ; []\U/stmry/m/n/10.95 [2\OML/cmm/m/it/10.95 ; \U/stmry/m/n/
10.95 3\OML/cmm/m/it/10.95 ; \U/stmry/m/n/10.95 6]\OML/cmm/m/it/10.95 ; []\U/st
mry/m/n/10.95 [2\OML/cmm/m/it/10.95 ; \U/stmry/m/n/10.95 4\OML/cmm/m/it/10.95 ;
 \U/stmry/m/n/10.95 5]\OML/cmm/m/it/10.95 ; []\U/stmry/m/n/10.95 [2\OML/cmm/m/i
t/10.95 ; \U/stmry/m/n/10.95 4\OML/cmm/m/it/10.95 ; \U/stmry/m/n/10.95 6]] ]
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[]\T1/pcr/m/n/10.95 norm($\OML/cmm/m/it/10.95 L$\T1/pcr/m/n/10.95 )[] \T1/ptm/m
/n/10.95 or \T1/pcr/m/n/10.95 l2norm($\OML/cmm/m/it/10.95 L$\T1/pcr/m/n/10.95 )
[] \T1/ptm/m/n/10.95 re-turns the $\OML/cmm/m/it/10.95 `[]$ \T1/ptm/m/n/10.95 n
orm of $\OML/cmm/m/it/10.95 L$\T1/ptm/m/n/10.95 ; namely $[]$. 
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[]\T1/ptm/b/n/12 Statistics on lists: \T1/pcr/b/n/12 mean variance stddev stdde
vp median
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[]\T1/pcr/m/n/10.95 variance($\OML/cmm/m/it/10.95 L$\T1/pcr/m/n/10.95 ) \T1/ptm
/m/n/10.95 com-putes the vari-ance of $\OML/cmm/m/it/10.95 L$\T1/ptm/m/n/10.95 
, which is the square of \T1/pcr/m/n/10.95 stddevp($\OML/cmm/m/it/10.95 L$\T1/p
cr/m/n/10.95 )\T1/ptm/m/n/10.95 . 
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[]\T1/ptm/m/n/10.95 If \T1/pcr/m/n/10.95 T \T1/ptm/m/n/10.95 is a sym-bolic var
i-able then the as-sign-ment \T1/pcr/m/n/10.95 T(\T1/ptm/m/it/10.95 index_name\
T1/pcr/m/n/10.95 :=\T1/ptm/m/it/10.95 element_value
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[]\T1/pcr/m/n/10.95 vandermonde($\OML/cmm/m/it/10.95 X$\T1/pcr/m/n/10.95 ) \T1/
ptm/m/n/10.95 re-turns the cor-re-spond-ing Van-der-monde ma-trix; namely,
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 \T1/pcr/m/n/10.95 border($\OML/cmm/m/it/10.95 A; b$\T1/pcr/m/n/10.95 )=tran([o
p(tran($\OML/cmm/m/it/10.95 A$\T1/pcr/m/n/10.95 )),$\OML/cmm/m/it/10.95 b$\T1/p
cr/m/n/10.95 ])=tran(append(tran($\OML/cmm/m/it/10.95 A$\T1/pcr/m/n/10.95 ),$\O
ML/cmm/m/it/10.95 b$\T1/pcr/m/n/10.95 )) 
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[]\T1/ptm/b/n/12 Creating a ma-trix with a for-mula or func-tion: \T1/pcr/b/n/1
2 makemat matrix
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[]\T1/pcr/m/n/10.95 subMat($\OML/cmm/m/it/10.95 A \OMS/cmsy/m/n/10.95 h\OML/cmm
/m/it/10.95 ; r[]; c[]; r[]; c[]\OMS/cmsy/m/n/10.95 i$\T1/pcr/m/n/10.95 ) \T1/p
tm/m/n/10.95 re-turns the sub-matrix of $\OML/cmm/m/it/10.95 A$ \T1/ptm/m/n/10.
95 from po-si-tion $\U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 r[]; c[]\U/stmry/m/
n/10.95 )$
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[]\T1/ptm/b/n/12 Cumulated sum of el-e-ments of each col-umn of a ma-trix: \T1/
pcr/b/n/12 cumSum
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[]\T1/ptm/b/n/12 Multiplication of el-e-ments of each col-umn of a ma-trix: \T1
/pcr/b/n/12 product
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\T1/ptm/m/n/10.95 The \T1/pcr/m/n/10.95 hadamard \T1/ptm/m/n/10.95 com-mand can
 find the Hadamard prod-uct of two ma-tri-ces; namely,
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[]\T1/ptm/b/n/12 Statistics func-tions act-ing on col-umn ma-tri-ces: \T1/pcr/b
/n/12 mean stddev
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 \T1/pcr/m/n/10.95 [3,4,2,5,6,0,1],[4,2,5,6,0,1,3],[2,5,6,0,1,3,4]],0.25) 
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[]\T1/ptm/b/n/12 Characteristic poly-no-mial us-ing Hes-sen-berg al-go-rithm: \
T1/pcr/b/n/12 pcar_hessenberg
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[]\T1/pcr/m/n/10.95 cholesky($\OML/cmm/m/it/10.95 M$\T1/pcr/m/n/10.95 ) \T1/ptm
/m/n/10.95 re-turns a sym-bolic or nu-meric ma-trix $\OML/cmm/m/it/10.95 P$ \T1
/ptm/m/n/10.95 given by the Cholesky
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cmm/m/it/10.95 A$ \T1/ptm/m/n/10.95 is not a sym-met-ric ma-trix, \T1/pcr/m/n/1
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\T1/ptm/m/n/10.95 the per-mu-ta-tion $\OML/cmm/m/it/10.95 p$\T1/ptm/m/n/10.95 .
 You can get the per-mu-ta-tion ma-trix from $\OML/cmm/m/it/10.95 p$ \T1/ptm/m/
n/10.95 by $\OML/cmm/m/it/10.95 P$\T1/pcr/m/n/10.95 :=permu2mat($\OML/cmm/m/it/
10.95 p$\T1/pcr/m/n/10.95 )
 []

[510] [511] [512]
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 \T1/pcr/m/n/10.95 evalf(sqrt(eigenvals([[1,2],[3,4]]*transpose([[1,2],[3,4]]))
)) 
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[] \OML/cmm/m/it/10.95 ; [] ; [] ; []
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[517]
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[]\T1/ptm/b/n/12 The op-er-a-tor norm of a ma-trix: \T1/pcr/b/n/12 matrix_norm 
l1norm l2norm
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[518]
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\T1/pcr/m/n/10.95 matrix_norm($\OML/cmm/m/it/10.95 A$\T1/pcr/m/n/10.95 ,1) \T1/
ptm/m/n/10.95 is the same as \T1/pcr/m/n/10.95 colnorm($\OML/cmm/m/it/10.95 A$\
T1/pcr/m/n/10.95 ) \T1/ptm/m/n/10.95 and \T1/pcr/m/n/10.95 l1norm($\OML/cmm/m/i
t/10.95 A$\T1/pcr/m/n/10.95 )\T1/ptm/m/n/10.95 . 
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\T1/pcr/m/n/10.95 matrix_norm($\OML/cmm/m/it/10.95 A$\T1/pcr/m/n/10.95 ,inf) \T
1/ptm/m/n/10.95 is the same as \T1/pcr/m/n/10.95 rownorm($\OML/cmm/m/it/10.95 A
$\T1/pcr/m/n/10.95 ) \T1/ptm/m/n/10.95 and \T1/pcr/m/n/10.95 linfnorm($\OML/cmm
/m/it/10.95 A$\T1/pcr/m/n/10.95 )\T1/ptm/m/n/10.95 . 
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[519] [520] [521] [522] [523]

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[524] [525] [526]
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 \T1/pcr/m/n/10.95 simplex_reduce([[-1,-1,0,1,1,0,1],[0,1,-1,1,0,1,3],[0,0,0,0,
1,1,0]]) 
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 \T1/pcr/m/n/10.95 simplex_reduce([[-1/2,0,-1/2,1,2],[1/2,1,-1/2,0,1],[2,3,-1,1
,0]]) 
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[527]
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[]\T1/pcr/m/n/10.95 assume\T1/ptm/m/n/10.95 , whose value can be one of \T1/pcr
/m/n/10.95 lp_nonnegative\T1/ptm/m/n/10.95 , \T1/pcr/m/n/10.95 lp_integer
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\T1/ptm/m/n/10.95 (\T1/pcr/m/n/10.95 integer\T1/ptm/m/n/10.95 ), \T1/pcr/m/n/10
.95 lp_binary \T1/ptm/m/n/10.95 or \T1/pcr/m/n/10.95 lp_nonnegint \T1/ptm/m/n/1
0.95 (or \T1/pcr/m/n/10.95 nonnegint\T1/ptm/m/n/10.95 )
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\T1/ptm/m/n/10.95 ti-fiers or in-dices (of in-te-ger vari-ables) (by de-fault, 
\T1/pcr/m/n/10.95 lp_integervariables=[]\T1/ptm/m/n/10.95 ). 
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\T1/ptm/m/n/10.95 fiers or in-dices (of bi-nary vari-ables) (by de-fault, \T1/p
cr/m/n/10.95 lp_binaryvariables=[]\T1/ptm/m/n/10.95 ). 
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Overfull \hbox (44.91794pt too wide) in paragraph at lines 40730--40733
\T1/pcr/m/n/10.95 false \T1/ptm/m/n/10.95 giv-ing the ob-jec-tive di-rec-tion (
by de-fault, \T1/pcr/m/n/10.95 lp_maximize=false\T1/ptm/m/n/10.95 ). 
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Overfull \hbox (31.69139pt too wide) in paragraph at lines 40733--40737
[]\T1/pcr/m/n/10.95 lp_method\T1/ptm/m/n/10.95 , whose value can be one of \T1/
pcr/m/n/10.95 exact\T1/ptm/m/n/10.95 , \T1/pcr/m/n/10.95 float\T1/ptm/m/n/10.95
 , \T1/pcr/m/n/10.95 lp_simplex
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\T1/ptm/m/n/10.95 or \T1/pcr/m/n/10.95 lp_interiorpoint\T1/ptm/m/n/10.95 , giv-
ing the solver type (by de-fault, \T1/pcr/m/n/10.95 lp_method=lp_simplex\T1/ptm
/m/n/10.95 ). 
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[]\T1/pcr/m/n/10.95 lp_nodelimit\T1/ptm/m/n/10.95 , whose value can be a pos-i-
tive in-te-ger or \T1/pcr/m/n/10.95 unlimited\T1/ptm/m/n/10.95 ,
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\T1/ptm/m/n/10.95 the max-i-mum solv-ing time in mil-lisec-onds (by de-fault: \
T1/pcr/m/n/10.95 lp_timelimit=unlimited\T1/ptm/m/n/10.95 ). 
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Overfull \hbox (55.56084pt too wide) in paragraph at lines 40754--40757
\T1/ptm/m/n/10.95 ing the rel-a-tive in-te-gral-ity gap thresh-old (by de-fault
, \T1/pcr/m/n/10.95 lp_gaptolerance=0\T1/ptm/m/n/10.95 ). 
 []

[528]
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[]\T1/pcr/m/n/10.95 lp_nodeselect\T1/ptm/m/n/10.95 , whose value can be one of 
\T1/pcr/m/n/10.95 lp_depthfirst\T1/ptm/m/n/10.95 ,
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\T1/pcr/m/n/10.95 lp_breadthfirst\T1/ptm/m/n/10.95 , \T1/pcr/m/n/10.95 lp_hybri
d \T1/ptm/m/n/10.95 or \T1/pcr/m/n/10.95 lp_bestprojection\T1/ptm/m/n/10.95 ,
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\T1/ptm/m/n/10.95 giv-ing the branch-ing node se-lec-tion strat-egy, (by de-fau
lt: \T1/pcr/m/n/10.95 lp_nodeselect=lp_hybrid\T1/ptm/m/n/10.95 ). 
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[]\T1/pcr/m/n/10.95 lp_varselect\T1/ptm/m/n/10.95 , whose value can be one of \
T1/pcr/m/n/10.95 lp_firstfractional\T1/ptm/m/n/10.95 ,
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\T1/pcr/m/n/10.95 lp_mostfractional \T1/ptm/m/n/10.95 or \T1/pcr/m/n/10.95 lp_p
seudocost \T1/ptm/m/n/10.95 giv-ing the branch-
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\T1/ptm/m/n/10.95 ing vari-able se-lec-tion strat-egy (by de-fault: \T1/pcr/m/n
/10.95 lp_varselect=lp_pseudocost\T1/ptm/m/n/10.95 ). 
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[529]
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 \T1/pcr/m/n/10.95 lpsolve(-x-y,[y<=3x+1/2,y<=-5x+2],assume=lp_nonnegative) 
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[530]
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 \T1/pcr/m/n/10.95 lpsolve(8x1+11x2+6x3+4x4,[5x1+7x2+4x3+3x4<=14],assume=lp_bin
ary,maximize) 
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[531]
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[532] [533]
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\T1/ptm/m/n/10.95 fore an op-ti-mal so-lu-tion is found. When \T1/pcr/m/n/10.95
 lp_nodeselect \T1/ptm/m/n/10.95 is set to \T1/pcr/m/n/10.95 lp_breadthfirst
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\T1/ptm/m/n/10.95 the solver needs to ex-am-ine only about 20000 sub-prob-lems,
 but when set to \T1/pcr/m/n/10.95 lp_hybrid
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 []\T1/ptm/b/n/10.95 Displaying de-tailed out-put.[] \T1/ptm/m/n/10.95 By typ-i
ng \T1/pcr/m/n/10.95 lp_verbose=true \T1/ptm/m/n/10.95 or sim-ply \T1/pcr/m/n/1
0.95 lp_verbose
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[534]
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 \T1/pcr/m/n/10.95 lpsolve(x1+x2,[1867*x1+1913*x2=3618894],assume=nonnegint,lp_
verbose=true) 
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[]\T1/pcr/m/n/10.95 Applying branch&bound method to find integer feasible solut
ions...[] 
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[535]
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\T1/ptm/m/n/10.95 solver by in-clud-ing \T1/pcr/m/n/10.95 lp_method=lp_interior
point \T1/ptm/m/n/10.95 af-ter \T1/pcr/m/n/10.95 assume=lp_nonnegative\T1/ptm/m
/n/10.95 :
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[536]
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[]\T1/ptm/m/n/10.95 For ex-am-ple, as-sume that \T1/pcr/m/n/10.95 somefile.lp \
T1/ptm/m/n/10.95 file is stored in di-rec-tory \T1/pcr/m/n/10.95 /path/to/file
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[]\T1/pcr/m/n/10.95 Reading problem data from '/path/to/file/somefile.lp'...[] 

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[][][]\T1/ptm/m/n/9 See \T1/pcr/m/n/9 https://www.gnu.org/software/glpk/ \T1/pt
m/m/n/9 for in-stalling GLPK in Linux or
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[537]
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\T1/ptm/m/n/10.95 num-ber of units that can be de-liv-ered from the $\OML/cmm/m
/it/10.95 k$\T1/ptm/m/n/10.95 th source for $\OML/cmm/m/it/10.95 k \U/stmry/m/n
/10.95 = 1\OML/cmm/m/it/10.95 ; \U/stmry/m/n/10.95 2\OML/cmm/m/it/10.95 ; [] ; 
m$\T1/ptm/m/n/10.95 , 
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[538]
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[539]
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[]\T1/ptm/m/n/10.95 Optionally, \T1/ptm/m/it/10.95 lo-ca-tion\T1/ptm/m/n/10.95 
, a key-word which may be \T1/pcr/m/n/10.95 coordinates\T1/ptm/m/n/10.95 , \T1/
pcr/m/n/10.95 locus
 []

[540] [541] [542] [543]
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[]\T1/pcr/m/n/10.95 maximize=\T1/ptm/m/it/10.95 bool\T1/ptm/m/n/10.95 , where \
T1/ptm/m/it/10.95 bool \T1/ptm/m/n/10.95 can be \T1/pcr/m/n/10.95 true \T1/ptm/
m/n/10.95 or \T1/pcr/m/n/10.95 false\T1/ptm/m/n/10.95 . Just \T1/pcr/m/n/10.95 
maximize
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\T1/ptm/m/n/10.95 is equiv-a-lent to \T1/pcr/m/n/10.95 maximize=true\T1/ptm/m/n
/10.95 . (By de-fault, \T1/pcr/m/n/10.95 maximize=false\T1/ptm/m/n/10.95 .) 
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[544]
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[]
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 \T1/pcr/m/n/10.95 nlpsolve(-x1*x2*x3,[72-x1-2x2-2x3>=0],x1=0..20,x2=0..11,x3=0
..42) 
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 \T1/pcr/m/n/10.95 conjugate_gradient([[2,1],[1,5]],[1,0],[0.55,-0.11],1e-2) 
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 \T1/pcr/m/n/10.95 conjugate_gradient([[2,1],[1,5]],[1,0],[0.55,-0.11],1e-10) 
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z-18) 
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[558]
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\T1/ptm/m/n/10.95 ing the col-umn vec-tor $\OML/cmm/m/it/10.95 b$ \T1/ptm/m/n/1
0.95 to the right of the ma-trix $\OML/cmm/m/it/10.95 A$ \T1/ptm/m/n/10.95 (as 
with \T1/pcr/m/n/10.95 border($\OML/cmm/m/it/10.95 A$\T1/pcr/m/n/10.95 ,tran($\
OML/cmm/m/it/10.95 b$\T1/pcr/m/n/10.95 ))\T1/ptm/m/n/10.95 ),
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[566]
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[]\T1/ptm/b/n/12 Solving a lin-ear sys-tem us-ing the Ja-cobi it-er-a-tion meth
od: \T1/pcr/b/n/12 jacobi_linsolve
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[567]
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[]\T1/ptm/b/n/12 Solving a lin-ear sys-tem us-ing the Gauss-Seidel it-er-a-tion
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[570]
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\T1/ptm/m/n/10.95 (Sec-tion [][]8.18[][] p.[][]636[][]), \T1/pcr/m/n/10.95 plot
ode \T1/ptm/m/n/10.95 (Sec-tion [][]8.19[][] p.[][]637[][]) and \T1/pcr/m/n/10.
95 interactive_plotode
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[571]
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 \T1/pcr/m/n/10.95 desolve([diff(y(t),t$2)+2*diff(y(t),t)+y(t),y(0)=1,y'(0)=0],
y(t)) 
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 \T1/pcr/m/n/10.95 desolve([diff(y(t),t$2)+2*diff(y(t),t)+y(t),y(0)=1,y'(0)=0],
t,y) 
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[572]
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\T1/ptm/m/n/10.95 where $\OML/cmm/m/it/10.95 a$\T1/ptm/m/n/10.95 , $\OML/cmm/m/
it/10.95 b$ \T1/ptm/m/n/10.95 and $\OML/cmm/m/it/10.95 c$ \T1/ptm/m/n/10.95 are
 ra-tio-nal func-tions of the in-de-pen-dent vari-able. \T1/pcr/m/n/10.95 kovac
icsols
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Overfull \hbox (4.92355pt too wide) in paragraph at lines 44559--44567
[]\T1/pcr/m/n/10.95 kovacicsols(\T1/ptm/m/it/10.95 kode$ \OMS/cmsy/m/n/10.95 h\
OML/cmm/m/it/10.95 x; y\OMS/cmsy/m/n/10.95 i$\T1/pcr/m/n/10.95 ) \T1/ptm/m/n/10
.95 re-turns a Li-ou-vil-lian so-lu-tion of equa-tion [].
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[581]
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 \T1/pcr/m/n/10.95 kovacicsols(x''+3*(t^^B2-t+1)/(16*(t-1)^^B2*t^^B2)*x,t,x) 
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\OMS/cmsy/m/n/10.95 ^^@\OML/cmm/m/it/10.95 omega[][] \OMS/cmsy/m/n/10.95 ^^C \O
ML/cmm/m/it/10.95 x[] \OMS/cmsy/m/n/10.95 ^^C \U/stmry/m/n/10.95 (\OML/cmm/m/it
/10.95 x \OMS/cmsy/m/n/10.95 ^^@ \U/stmry/m/n/10.95 1)[] + \OML/cmm/m/it/10.95 
omega[][] \OMS/cmsy/m/n/10.95 ^^C \OML/cmm/m/it/10.95 x[] \OMS/cmsy/m/n/10.95 ^
^C \U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 x \OMS/cmsy/m/n/10.95 ^^@ \U/stmry/m
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m/m/it/10.95 =\U/stmry/m/n/10.95 3 \OMS/cmsy/m/n/10.95 ^^@ \OML/cmm/m/it/10.95 
omega[][] \OMS/cmsy/m/n/10.95 ^^C \OML/cmm/m/it/10.95 x[] \OMS/cmsy/m/n/10.95 ^
^C \U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 x \OMS/cmsy/m/n/10.95 ^^@ \U/stmry/m
/n/10.95 1)[] \OMS/cmsy/m/n/10.95 ^^C \U/stmry/m/n/10.95 (48 \OMS/cmsy/m/n/10.9
5 ^^C \OML/cmm/m/it/10.95 x[] \OMS/cmsy/m/n/10.95 ^^@ \U/stmry/m/n/10.95 41 \OM
S/cmsy/m/n/10.95 ^^C \OML/cmm/m/it/10.95 x \U/stmry/m/n/10.95 + 9)\OML/cmm/m/it
/10.95 =\U/stmry/m/n/10.95 24 + \OML/cmm/m/it/10.95 omega[] \OMS/cmsy/m/n/10.95
 ^^C \OML/cmm/m/it/10.95 x \OMS/cmsy/m/n/10.95 ^^C \U/stmry/m/n/10.95 (\OML/cmm
/m/it/10.95 x \OMS/cmsy/m/n/10.95 ^^@ \U/stmry/m/n/10.95 1) \OMS/cmsy/m/n/10.95
 ^^C \U/stmry/m/n/10.95 (320 \OMS/cmsy/m/n/10.95 ^^C \OML/cmm/m/it/10.95 x[] \O
MS/cmsy/m/n/10.95 ^^@ \U/stmry/m/n/10.95 409 \OMS/cmsy/m/n/10.95 ^^C \OML/cmm/m
/it/10.95 x[] \U/stmry/m/n/10.95 + 180 \OMS/cmsy/m/n/10.95 ^^C \OML/cmm/m/it/10
.95 x \OMS/cmsy/m/n/10.95 ^^@ \U/stmry/m/n/10.95 27)\OML/cmm/m/it/10.95 =\U/stm
ry/m/n/10.95 432 + (\OMS/cmsy/m/n/10.95 ^^@\U/stmry/m/n/10.95 2048 \OMS/cmsy/m/
n/10.95 ^^C \OML/cmm/m/it/10.95 x[] \U/stmry/m/n/10.95 + 3484 \OMS/cmsy/m/n/10.
95 ^^C \OML/cmm/m/it/10.95 x[] \OMS/cmsy/m/n/10.95 ^^@ \U/stmry/m/n/10.95 2313 
\OMS/cmsy/m/n/10.95 ^^C \OML/cmm/m/it/10.95 x[] \U/stmry/m/n/10.95 + 702 \OMS/c
msy/m/n/10.95 ^^C \OML/cmm/m/it/10.95 x \OMS/cmsy/m/n/10.95 ^^@ \U/stmry/m/n/10
.95 81)\OML/cmm/m/it/10.95 =\U/stmry/m/n/10.95 20736
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[]\OML/cmm/m/it/10.95 ![][] [] \OMS/cmsy/m/n/10.95 ^^@ []\OML/cmm/m/it/10.95 ![
][]t [] [] \OMS/cmsy/m/n/10.95 ^^@ []\OML/cmm/m/it/10.95 ![][]t[] [] [] \U/stmr
y/m/n/10.95 + []\OML/cmm/m/it/10.95 ![]t[] [] [][] [] \OMS/cmsy/m/n/10.95 ^^@ \
OML/cmm/m/it/10.95 t[] [][] [] []
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\T1/ptm/m/n/10.95 de-fault \T1/pcr/m/n/10.95 DIGITS\T1/ptm/m/n/10.95 , which it
-self has a de-fault value of 12; see Sec-tion [][]3.5.1[][]
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Chapter 7.

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[]\T1/ptm/b/n/14.4 Computing the cur-va-ture and re-lated val-ues: \T1/pcr/b/n/
14.4 curvature
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\T1/ptm/m/sl/10.95 7.2.  COMPUTING THE CURVATURE AND RELATED VALUES: \T1/pcr/m/
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Chapter 8.

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wFunc
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\T1/ptm/m/sl/10.95 8.4.  GRAPH OF A FUNCTION: \T1/pcr/m/sl/10.95 PLOTFUNC FUNCP
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[]\T1/ptm/m/n/10.95 Optionally, \T1/ptm/m/it/10.95 xstep\T1/ptm/m/n/10.95 , whi
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[]\T1/ptm/b/n/14.4 Graph of in-equal-i-ties with 2 vari-ables: \T1/pcr/b/n/14.4
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8[][]
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[]\T1/ptm/m/n/10.95 If you want to draw the sur-face in 3-d rep-re-sen-ta-tion,
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[]\T1/pcr/m/n/10.95 plotfield($\OML/cmm/m/it/10.95 V$\T1/pcr/m/n/10.95 ,\T1/ptm
/m/it/10.95 vars$ \OMS/cmsy/m/n/10.95 h\OML/cmm/m/it/10.95 ;$\T1/pcr/m/n/10.95 
xstep=$\OML/cmm/m/it/10.95 n;$\T1/pcr/m/n/10.95 ystep=$\OML/cmm/m/it/10.95 m\OM
S/cmsy/m/n/10.95 i$\T1/pcr/m/n/10.95 ) \T1/ptm/m/n/10.95 draws the vec-tor fiel
d
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[]\T1/ptm/b/n/14.4 Plotting a so-lu-tion of a dif-fer-en-tial equa-tion: \T1/pc
r/b/n/14.4 plotode
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\T1/ptm/m/sl/10.95 8.19.  PLOTTING A SOLUTION OF A DIFFERENTIAL EQUATION: \T1/p
cr/m/sl/10.95 PLOTODE ODEPLOT    \T1/ptm/m/n/10.95 637
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\T1/ptm/b/n/14.4 equa-tion: \T1/pcr/b/n/14.4 interactive_plotode interactive_od
eplot
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\T1/ptm/m/sl/10.95 8.20.  INTERACTIVE PLOTTING OF SOLUTIONS OF A DIFFERENTIAL E
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m/n/10.95 639
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con-
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[]\T1/ptm/b/n/12 Animation of a se-quence of graphic ob-jects: \T1/pcr/b/n/12 a
nimation
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Chapter 9.

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[]\T1/ptm/b/n/12 The pop-u-la-tion stan-dard de-vi-a-tion: \T1/pcr/b/n/12 stdde
vp stdDev
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[]\T1/ptm/m/n/10.95 Optionally, $\OML/cmm/m/it/10.95 a$ \T1/ptm/m/n/10.95 and $
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[]\T1/ptm/b/n/12 Cumulative fre-quen-cies: \T1/pcr/b/n/12 cumulated_frequencies
 frequences_cumulees
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[]\T1/ptm/b/n/12 Covariance and cor-re-la-tion: \T1/pcr/b/n/12 covariance corre
lation
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[]\T1/ptm/b/n/12 Polygonal paths: \T1/pcr/b/n/12 polygonplot ligne_polygonale l
inear_interpolate
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[]$\OML/cmm/m/it/10.95 M$\T1/ptm/m/n/10.95 , a two-row ma-trix con-sist-ing of 
the $\OML/cmm/m/it/10.95 x$\T1/ptm/m/n/10.95 -coordinates and the $\OML/cmm/m/i
t/10.95 y$\T1/ptm/m/n/10.95 -coordinates. 
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[]\T1/pcr/m/n/10.95 linear_interpolate($\OML/cmm/m/it/10.95 M; x[]; x[]; x[]$\T
1/pcr/m/n/10.95 ) \T1/ptm/m/n/10.95 re-turns a ma-trix with two
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[]\T1/ptm/b/n/12 Linear re-gres-sion: \T1/pcr/b/n/12 linear_regression linear_r
egression_plot
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[]\T1/ptm/b/n/12 Exponential re-gres-sion: \T1/pcr/b/n/12 exponential_regressio
n exponential_regression_plot
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[]\T1/ptm/b/n/12 Logarithmic re-gres-sion: \T1/pcr/b/n/12 logarithmic_regressio
n logarithmic_regression_plot
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\T1/ptm/m/n/10.95 so the best fit log-a-rith-mic curve will be $\OML/cmm/m/it/1
0.95 y \U/stmry/m/n/10.95 = 10\OML/cmm/m/it/10.95 :\U/stmry/m/n/10.95 150645000
2 [](\OML/cmm/m/it/10.95 x\U/stmry/m/n/10.95 ) \OMS/cmsy/m/n/10.95 ^^@ \U/stmry
/m/n/10.95 0\OML/cmm/m/it/10.95 :\U/stmry/m/n/10.95 564824055818$\T1/ptm/m/n/10
.95 . 
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[]\T1/ptm/b/n/12 Power re-gres-sion: \T1/pcr/b/n/12 power_regression power_regr
ession_plot
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[]\T1/ptm/b/n/12 Polynomial re-gres-sion: \T1/pcr/b/n/12 polynomial_regression 
polynomial_regression_plot
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[]\T1/pcr/m/n/10.95 polynomial_regression(\T1/ptm/m/it/10.95 xcoords,ycoords,$\
OML/cmm/m/it/10.95 n$\T1/pcr/m/n/10.95 ) \T1/ptm/m/n/10.95 re-turns the list $\
U/stmry/m/n/10.95 [\OML/cmm/m/it/10.95 a[]; [] ; a[]\U/stmry/m/n/10.95 ]$
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[]\T1/ptm/b/n/12 Logistic re-gres-sion: \T1/pcr/b/n/12 logistic_regression logi
stic_regression_plot
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[]\T1/pcr/m/n/10.95 logistic_regression($\OML/cmm/m/it/10.95 L; x[]; y[]$\T1/pc
r/m/n/10.95 ) \T1/ptm/m/n/10.95 re-turns a list $\U/stmry/m/n/10.95 [\OML/cmm/m
/it/10.95 y; y[]; C; y[]; x[]; R; Y\U/stmry/m/n/10.95 ]$
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[]   \T1/pcr/m/n/10.95 Correlation 0.780548607383, Estimated total P=-0.1559139
78495[] 
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[]   \T1/pcr/m/n/10.95 Returning estimated Pcumul, Pinstant, Ptotal, Pinstantma
x, tmax, R[] 
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[]\T1/pcr/m/n/10.95 logistic_regression_plot($\OML/cmm/m/it/10.95 L; x[]; y[]$\
T1/pcr/m/n/10.95 ) \T1/ptm/m/n/10.95 draws the best fit lo-gis-tic equa-
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[676]
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[]\T1/ptm/b/n/12 Producing uni-formly dis-tributed ran-dom num-bers: \T1/pcr/b/
n/12 rand random
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[]\T1/ptm/m/n/10.95 Another way to get a ran-dom in-te-ger in a spec-i-fied in-
ter-val is with the \T1/pcr/m/n/10.95 randint
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1/pcr/b/n/12 randpoisson
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1/pcr/b/n/12 randstudent
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[]\T1/ptm/b/n/12 Producing ran-dom num-bers with the $\OML/cmm/m/it/12 ^^_[]$ \
T1/ptm/b/n/12 dis-tri-bu-tion: \T1/pcr/b/n/12 randchisquare
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\T1/ptm/m/n/10.95 The \T1/pcr/m/n/10.95 randfisher \T1/ptm/m/n/10.95 com-mand f
inds ran-dom num-bers cho-sen ac-cord-ing to the Fisher-
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\T1/ptm/m/n/10.95 The \T1/pcr/m/n/10.95 randgammad \T1/ptm/m/n/10.95 com-mand f
inds ran-dom num-bers cho-sen ac-cord-ing to the gamma
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[]\T1/ptm/b/n/12 Producing ran-dom num-bers with the beta dis-tri-bu-tion: \T1/
pcr/b/n/12 randbetad
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[]\T1/pcr/m/n/10.95 'binomial($\OML/cmm/m/it/10.95 n$\T1/pcr/m/n/10.95 ,$\OML/c
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/n/10.95 ,
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\T1/ptm/m/n/10.95 for a bi-no-mial dis-tri-bu-tion (see Sec-tion [][]9.4.3[][] 
p.[][]698[][] and Sec-tion [][]9.3.3[][]
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[]\T1/pcr/m/n/10.95 'multinomial($\OML/cmm/m/it/10.95 P$\T1/pcr/m/n/10.95 ,$\OM
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\T1/ptm/m/n/10.95 for a Pois-son dis-tri-bu-tion (see Sec-tion [][]9.4.6[][] p.
[][]703[][] and Sec-tion [][]9.3.5[][]
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[]\T1/pcr/m/n/10.95 'normald($\OML/cmm/m/it/10.95 ^^V$\T1/pcr/m/n/10.95 ,$\OML/
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[][]\T1/ptm/b/n/10.95 The prob-a-bil-ity den-sity func-tion for the uni-form di
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[][]\T1/ptm/b/n/10.95 The prob-a-bil-ity den-sity func-tion for the neg-a-tive 
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[]\U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 x \OMS/cmsy/m/n/10.95 ^^T \OML/cmm/m/
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\T1/ptm/m/n/10.95 bu-tion with $\OML/cmm/m/it/10.95 n$ \T1/ptm/m/n/10.95 de-gre
es of free-dom; namely, the value of $\OML/cmm/m/it/10.95 x$ \T1/ptm/m/n/10.95 
for which $[]\U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 X \OMS/cmsy/m/n/10.95 ^^T
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[][]\T1/ptm/b/n/10.95 The cu-mu-la-tive dis-tri-bu-tion func-tion for the $\OML
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\T1/ptm/m/n/10.95 bu-tion with $\OML/cmm/m/it/10.95 n$ \T1/ptm/m/n/10.95 de-gre
es of free-dom; namely, the value of $\OML/cmm/m/it/10.95 x$ \T1/ptm/m/n/10.95 
for which $[]\U/stmry/m/n/10.95 (\OML/cmm/m/it/10.95 X \OMS/cmsy/m/n/10.95 ^^T
 []


Overfull \hbox (7.96387pt too wide) in paragraph at lines 52700--52703
\T1/ptm/m/n/10.95 The \T1/pcr/m/n/10.95 UTPC \T1/ptm/m/n/10.95 (the Up-per Tail
 Prob-a-bil-ity - Chi-square dis-tri-bu-tion) com-putes $[]\U/stmry/m/n/10.95 (
\OML/cmm/m/it/10.95 X >
 []


Overfull \hbox (3.17429pt too wide) in paragraph at lines 52735--52735
[][]\T1/ptm/b/n/10.95 The prob-a-bil-ity den-sity func-tion for the Fisher-Snéd
écor dis-tri-bu-tion: \T1/pcr/b/n/10.95 fisher
 []

[711]
Overfull \hbox (38.48785pt too wide) in paragraph at lines 52838--52838
[][]\T1/ptm/b/n/10.95 The in-verse dis-tri-bu-tion func-tion for the Fisher-Sné
décor dis-tri-bu-tion: \T1/pcr/b/n/10.95 fisher_icdf
 []

[712]
Overfull \hbox (1.20322pt too wide) in paragraph at lines 52869--52869
[][]\T1/ptm/b/n/10.95 The up-per tail cu-mu-la-tive func-tion for the Fisher-Sn
édécor dis-tri-bu-tion: \T1/pcr/b/n/10.95 UTPF
 []

[713]
Overfull \hbox (8.9346pt too wide) in paragraph at lines 52940--52940
[][]\T1/ptm/b/n/10.95 The cu-mu-la-tive dis-tri-bu-tion func-tion for the gamma
 dis-tri-bu-tion: \T1/pcr/b/n/10.95 gammad_cdf
 []

[714] [715] [716]
Overfull \hbox (35.5975pt too wide) in paragraph at lines 53208--53208
[][]\T1/ptm/b/n/10.95 The cu-mu-la-tive dis-tri-bu-tion func-tion of the ge-o-m
et-ric dis-tri-bu-tion: \T1/pcr/b/n/10.95 geometric_cdf
 []

[717]
Overfull \hbox (28.1735pt too wide) in paragraph at lines 53254--53254
[][]\T1/ptm/b/n/10.95 The in-verse dis-tri-bu-tion func-tion for the ge-o-met-r
ic dis-tri-bu-tion: \T1/pcr/b/n/10.95 geometric_icdf
 []


Overfull \hbox (10.74156pt too wide) in paragraph at lines 53288--53288
[][]\T1/ptm/b/n/10.95 The prob-a-bil-ity den-sity func-tion for the Cauchy dis-
tri-bu-tion: \T1/pcr/b/n/10.95 cauchy cauchyd
 []


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 []

[718]
Overfull \hbox (9.99666pt too wide) in paragraph at lines 53333--53333
[][]\T1/ptm/b/n/10.95 The cu-mu-la-tive dis-tri-bu-tion func-tion for the Cauch
y dis-tri-bu-tion: \T1/pcr/b/n/10.95 cauchy_cdf
 []

[719]
Overfull \hbox (14.32993pt too wide) in paragraph at lines 53413--53417
\T1/ptm/m/n/10.95 dis-tri-bu-tion with pa-ram-e-ters $\OML/cmm/m/it/10.95 a$ \T
1/ptm/m/n/10.95 and $\OML/cmm/m/it/10.95 b$\T1/ptm/m/n/10.95 ; namely, the valu
e of $\OML/cmm/m/it/10.95 x$ \T1/ptm/m/n/10.95 for which $[]\U/stmry/m/n/10.95 
(\OML/cmm/m/it/10.95 X \OMS/cmsy/m/n/10.95 ^^T
 []


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 []


Overfull \hbox (13.93849pt too wide) in paragraph at lines 53436--53436
[][]\T1/ptm/b/n/10.95 The prob-a-bil-ity den-sity func-tion for the ex-po-nen-t
ial dis-tri-bu-tion: \T1/pcr/b/n/10.95 exponential
 []


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 []


Overfull \hbox (61.26404pt too wide) in paragraph at lines 53468--53468
[][]\T1/ptm/b/n/10.95 The cu-mu-la-tive dis-tri-bu-tion func-tion for the ex-po
-nen-tial dis-tri-bu-tion: \T1/pcr/b/n/10.95 exponential_cdf
 []

[720]
Overfull \hbox (49.25206pt too wide) in paragraph at lines 53513--53513
[][]\T1/ptm/b/n/10.95 The in-verse dis-tri-bu-tion func-tion for the ex-po-nen-
tial dis-tri-bu-tion: \T1/pcr/b/n/10.95 exponential_icdf
 []


Overfull \hbox (1.27173pt too wide) in paragraph at lines 53515--53517
\T1/ptm/m/n/10.95 The \T1/pcr/m/n/10.95 exponential_icdf \T1/ptm/m/n/10.95 (or 
\T1/pcr/m/n/10.95 exponentiald_icdf\T1/ptm/m/n/10.95 ) com-mand com-putes the
 []


Overfull \hbox (24.34123pt too wide) in paragraph at lines 53548--53548
[][]\T1/ptm/b/n/10.95 The prob-a-bil-ity den-sity func-tion for the Weibull dis
-tri-bu-tion: \T1/pcr/b/n/10.95 weibull weibulld
 []

[721]
Overfull \hbox (17.02634pt too wide) in paragraph at lines 53591--53591
[][]\T1/ptm/b/n/10.95 The cu-mu-la-tive dis-tri-bu-tion func-tion for the Weibu
ll dis-tri-bu-tion: \T1/pcr/b/n/10.95 weibull_cdf
 []

[722]
Overfull \hbox (5.01436pt too wide) in paragraph at lines 53656--53656
[][]\T1/ptm/b/n/10.95 The in-verse dis-tri-bu-tion func-tion for the Weibull di
s-tri-bu-tion: \T1/pcr/b/n/10.95 weibull_icdf
 []


Overfull \hbox (4.21315pt too wide) in paragraph at lines 53669--53673
[]\T1/pcr/m/n/10.95 weibull_icdf($\OML/cmm/m/it/10.95 k; ^^U\U/stmry/m/n/10.95 
[\OML/cmm/m/it/10.95 ; ^^R\U/stmry/m/n/10.95 ]\OML/cmm/m/it/10.95 ; h$\T1/pcr/m
/n/10.95 ) \T1/ptm/m/n/10.95 re-turns the in-verse dis-tri-bu-tion for the Weib
ull
 []

[723]
Overfull \hbox (4.64943pt too wide) in paragraph at lines 53700--53702
[]\T1/pcr/m/n/10.95 kolmogorovd($\OML/cmm/m/it/10.95 x$\T1/pcr/m/n/10.95 ) \T1/
ptm/m/n/10.95 re-turns the den-sity func-tion of the Kolmogorov-Smirnov
 []

[724]
Overfull \hbox (94.42455pt too wide) in paragraph at lines 53805--53806
 \T1/pcr/m/n/10.95 wilcoxons([1,3,4,5,7,8,8,12,15,17],[2,6,10,11,13,14,15,18,19
,20]) 
 []


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 \T1/pcr/m/n/10.95 wilcoxont([1,2,3,4,5,7,8,8,12,15,17],[2,6,10,11,13,14,15,18,
19,20]) 
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Overfull \hbox (9.01465pt too wide) in paragraph at lines 53863--53863
 []   \T1/pcr/m/n/10.95 Mann-Whitney 2-sample test, H0 same Median, H1 <>[] 
 []

[725]
Overfull \hbox (114.13452pt too wide) in paragraph at lines 53863--53863
 []   \T1/pcr/m/n/10.95 P-value 9055/176358 (0.0513444244094), alpha=0.05 H0 no
t rejected[] 
 []


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 []  \T1/pcr/m/n/10.95 \texttt{wilcoxont([1,3,4,5,7,8,8,12,15,17],[2,6,10,11,13
,14,15,18,19,20],0.3)}[] 
 []


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 []   \T1/pcr/m/n/10.95 Mann-Whitney 2-sample test, H0 same Median, H1 <>[] 
 []


Overfull \hbox (61.57458pt too wide) in paragraph at lines 53863--53863
 []   \T1/pcr/m/n/10.95 P-value 316/4199 (0.0752560133365), alpha=0.3 H0 reject
ed[] 
 []


Overfull \hbox (94.42455pt too wide) in paragraph at lines 53874--53874
 []   \T1/pcr/m/n/10.95 Wilcoxon statistic: 18, p-value: 0.375, confidence leve
l: 0.05[] 
 []


Overfull \hbox (4.94077pt too wide) in paragraph at lines 53878--53878
[]\T1/ptm/b/n/12 Moment gen-er-at-ing func-tions for prob-a-bil-ity dis-tri-bu-
tions: \T1/pcr/b/n/12 mgf
 []

[726] [727]
Overfull \hbox (1.33679pt too wide) in paragraph at lines 54033--54035
[]\T1/pcr/m/n/10.95 kernel_density \T1/ptm/m/n/10.95 takes one manda-tory ar-gu
-ment and an un-spec-i-fied num-
 []

[728]
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\T1/ptm/m/n/10.95 range end-points are not spec-i-fied, they are set to $\OML/c
mm/m/it/10.95 a \U/stmry/m/n/10.95 = [][] \OML/cmm/m/it/10.95 X[] \OMS/cmsy/m/n
/10.95 ^^@
 []

[729]
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[731]
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[733]
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 \T1/pcr/m/n/10.95 randmarkov([[0,1/2,0,1/2],[0,1,0,0],[1/4,1/4,1/4,1/4],[0,0,1
/2,1/2]],2,10) 
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[]\T1/pcr/m/n/10.95 randmatrix($\OML/cmm/m/it/10.95 v; i[]$\T1/pcr/m/n/10.95 ) 
\T1/ptm/m/n/10.95 re-turns a stochas-tic ma-trix with $\OML/cmm/m/it/10.95 p$ \
T1/ptm/m/n/10.95 re-cur-rent loops (given
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[]
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[734]
Overfull \hbox (114.13452pt too wide) in paragraph at lines 54450--54450
 []   \T1/pcr/m/n/10.95 Summary Z-Test null hypothesis H0 mu1=mu2, alt. hyp. H1
 mu1!=mu2.[] 
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 []   \T1/pcr/m/n/10.95 Test returns 0 if probability to observe data is less t
han 0.1[] 
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ility error)[] 
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 []   \T1/pcr/m/n/10.95 Test returns 1 otherwise (can not reject null hypothesi
s)[] 
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Overfull \hbox (173.26445pt too wide) in paragraph at lines 54450--54450
 []   \T1/pcr/m/n/10.95 alpha level 0.1, multiplier*stddev/sqrt(sample size)= 1
.64485*0.02/5.47723[] 
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 []   \T1/pcr/m/n/10.95 Summary Z-Test null hypothesis H0 mu1=mu2, alt. hyp. H1
 mu1<mu2.[] 
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han 0.05[] 
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1.64485*0.1/7.07107[] 
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[735]
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 mu1!=mu2.[] 
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han 0.1[] 
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s)[] 
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 []   \T1/pcr/m/n/10.95 Data mean mu1=10, population mean mu2=0.5, degrees of f
reedom 20[] 
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 []   \T1/pcr/m/n/10.95 alpha level 0.1, multiplier*stddev/sqrt(sample size)= 1
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Overfull \hbox (107.56453pt too wide) in paragraph at lines 54536--54536
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 mu1<mu2.[] 
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s)[] 
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 []   \T1/pcr/m/n/10.95 Data mean mu1=0.48, population mean mu2=0.5, degrees of
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 []   \T1/pcr/m/n/10.95 alpha level 0.05, multiplier*stddev/sqrt(sample size)= 
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[736]
Overfull \hbox (8.78218pt too wide) in paragraph at lines 54556--54562
[]\T1/ptm/m/it/10.95 params\T1/ptm/m/n/10.95 , the pa-ram-e-ters of the dis-tri
-bu-tion \T1/ptm/m/it/10.95 distr \T1/ptm/m/n/10.95 or the key-word \T1/pcr/m/n
/10.95 classes
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Overfull \hbox (87.85455pt too wide) in paragraph at lines 54587--54587
 []  \T1/pcr/m/n/10.95 Guessing data is the list of number of elements in each 
class,[] 
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Overfull \hbox (94.42455pt too wide) in paragraph at lines 54587--54587
 []  \T1/pcr/m/n/10.95 Sample adequation to a finite discrete probability distr
ibution[] 
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Overfull \hbox (409.78416pt too wide) in paragraph at lines 54587--54587
 []  \T1/pcr/m/n/10.95 reject adequation if superior to chisquare_icdf(1,0.95)=
3.84145882069 or chisquare_icdf(1,1-alpha) if alpha!=5%[] 
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ribution[] 
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Overfull \hbox (140.41449pt too wide) in paragraph at lines 54600--54600
 []   \T1/pcr/m/n/10.95 reject adequation if superior to chisquare_icdf(1,0.95)
=3.84145882069[] 
 []


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pha!=5%[] 
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 []   \T1/pcr/m/n/10.95 Binomial: estimating n and p from data 10 0.5055[] 
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[737]
Overfull \hbox (179.83444pt too wide) in paragraph at lines 54613--54613
 []   \T1/pcr/m/n/10.95 Sample adequation to binomial(10,0.5055,.), Chi2 test r
esult 7.77825189838,[] 
 []


Overfull \hbox (140.41449pt too wide) in paragraph at lines 54613--54613
 []   \T1/pcr/m/n/10.95 reject adequation if superior to chisquare_icdf(7,0.95)
=14.0671404493[] 
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pha!=5%[] 
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lt 125.617374161,[] 
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)=18.3070380533[] 
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lpha!=5%[] 
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 []        \T1/pcr/m/n/10.95 estimating mean and stddev from data -0.0034591975
2912 0.201708100832[] 
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 []   \T1/pcr/m/n/10.95 Sample adequation to normald_cdf(-0.00345919752912,0.20
1708100832,.),[] 
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Overfull \hbox (140.41449pt too wide) in paragraph at lines 54648--54648
 []   \T1/pcr/m/n/10.95 reject adequation if superior to chisquare_icdf(4,0.95)
=9.48772903678[] 
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\T1/ptm/m/n/10.95 In this last case, you are given the value of $\OML/cmm/m/it/
10.95 d[]$ \T1/ptm/m/n/10.95 of the statis-tic $\OML/cmm/m/it/10.95 D[] \U/stmr
y/m/n/10.95 = [][](\OML/cmm/m/it/10.95 n[] \OMS/cmsy/m/n/10.95 ^^@
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\OML/cmm/m/it/10.95 e[]\U/stmry/m/n/10.95 )\OML/cmm/m/it/10.95 =e[]$\T1/ptm/m/n
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ple classes for \T1/pcr/m/n/10.95 classes(L,-2,0.25)
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[738]
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[739] [740

]
Chapter 10.

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\T1/ptm/m/n/10.95 of the man-tissa of a cre-ated MPFR float is \T1/pcr/m/n/10.9
5 ceil(Digits*log(10)/log(2))\T1/ptm/m/n/10.95 . 
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[742]
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[]\T1/pcr/m/n/10.95 newton(\T1/ptm/m/it/10.95 ex,$ \OMS/cmsy/m/n/10.95 h$\T1/pt
m/m/it/10.95 ,var$\OML/cmm/m/it/10.95 ; a; ^^O;$\T1/ptm/m/it/10.95 nbiter$\OMS/
cmsy/m/n/10.95 h$\T1/pcr/m/n/10.95 ) \T1/ptm/m/n/10.95 re-turns an ap-prox-i-ma
te so-lu-tion of the equa-
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[]\T1/pcr/m/n/10.95 gaussquad(\T1/ptm/m/it/10.95 expr$ \OMS/cmsy/m/n/10.95 h\OM
L/cmm/m/it/10.95 ;$\T1/ptm/m/it/10.95 var$\OMS/cmsy/m/n/10.95 i\OML/cmm/m/it/10
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95 [\OML/cmm/m/it/10.95 x \U/stmry/m/n/10.95 = \OML/cmm/m/it/10.95 a::b; y\U/st
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Chapter 11.
[761

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Chapter 12.

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Chapter 13.

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\T1/ptm/m/n/10.95 Note that \T1/pcr/m/n/10.95 t \T1/ptm/m/n/10.95 must be a for
-mal vari-able, it may be nec-es-sary to purge it with \T1/pcr/m/n/10.95 purge(
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\T1/ptm/m/n/10.95 are used to in-ter-ac-tively find and dis-play the ap-pro-pri
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[]\T1/ptm/m/n/10.95 The \T1/pcr/m/n/10.95 distance2 \T1/ptm/m/n/10.95 com-mand 
finds the square of the dis-tance be-tween two points. 
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n/10.95 and $\OML/cmm/m/it/10.95 D$ \T1/ptm/m/n/10.95 are in har-monic di-vi-si
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Chapter 14.
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Chapter 15.

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[]\T1/ptm/m/n/10.95 As an ex-am-ple, sup-pose the fol-low-ing im-age is stored 
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[1011 <./xcas-image2.png>] [1012]
Chapter 16.

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[]\T1/pcr/m/n/10.95 // -*- compile-command: "g++ -g pgcd.cc -lgiac -lgmp" -*-[]
 
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[]\T1/pcr/m/n/10.95 gen pgcd(gen a,gen b){ gen q,r; for (;b!=0;){ r=irem(a,b,q)
; a=b; b=r;[] 
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[]\T1/pcr/m/n/10.95 int main(){ cout << "Enter 2 integers "; gen a,b; cin >> a 
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[]\T1/ptm/m/n/10.95 If \T1/pcr/m/n/10.95 g._CPLXptr \T1/ptm/m/n/10.95 is a com-
plex type \T1/pcr/m/n/10.95 complex\T1/ptm/m/n/10.95 , then \T1/pcr/m/n/10.95 g
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[]\T1/pcr/m/n/10.95 // -*- mode:C++ ; compile-command: "g++ -I.. -fPIC -DPIC -g
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d.so.0 -o \[] 
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[]\T1/pcr/m/n/10.95 #ifndef NO_NAMESPACE_GIAC namespace giac { #endif // ndef[]
 
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[]  \T1/pcr/m/n/10.95 gen monpgcd(const gen & a0,const gen & b0){ gen q,r,a=a0,
b=b0; for[] 
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[]  \T1/pcr/m/n/10.95 (;b!=0;){ r=irem(a,b,q); a=b; b=r; } return a; } gen _mon
pgcd(const[] 
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[]  \T1/pcr/m/n/10.95 (args._VECTptr->size()!=2)) setsizeerr(); vecteur &v=*arg
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[]  \T1/pcr/m/n/10.95 return monpgcd(v[0],v[1]); } const string _monpgcd_s("mon
pgcd");[] 
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[]  \T1/pcr/m/n/10.95 unary_function_eval __monpgcd(0,&_monpgcd,_monpgcd_s);[] 

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so.0 -o \[] 
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