INTRODUCTION ..................................... 1
1. ELEMENTARY PROCEDURES ......................... 3
1.1 Real vector and matrix — Initialization ............................ 3
A. inivec ......................................................... 3
B. inimat ......................................................... 3
C. inimatd ........................................................ 4
D. inisymd ........................................................ 4
E. inisymrow ...................................................... 5
1.2 Real vector and matrix — Duplication ............................... 6
A. dupvec ......................................................... 6
B. dupvecrow ...................................................... 6
C. duprowvec ...................................................... 7
D. dupveccol ...................................................... 7
E. dupcolvec ...................................................... 7
F. dupmat ......................................................... 8
1.3 Real vector and matrix — Multiplication ............................ 8
A. mulvec ......................................................... 8
B. mulrow ......................................................... 9
C. mulcol ......................................................... 9
D. colcst ........................................................ 10
E. rowcst ........................................................ 10
1.4 Real vector vector products ....................................... 10
A. vecvec ........................................................ 10
B. matvec ........................................................ 11
C. tamvec ........................................................ 12
D. matmat ........................................................ 12
E. tammat ........................................................ 13
F. mattam ........................................................ 13
G. seqvec ........................................................ 14
H. scaprdl ....................................................... 14
I. symmatvec ..................................................... 15
1.5 Real matrix vector products ....................................... 16
A. fulmatvec ..................................................... 16
B. fultamvec ..................................................... 16
C. fulsymmatvec .................................................. 17
D. resvec ........................................................ 17
E. symresvec ..................................................... 18
1.6 Real matrix matrix products ....................................... 18
A. hshvecmat ..................................................... 18
B. hshcolmat ..................................................... 19
C. hshrowmat ..................................................... 20
D. hshvectam ..................................................... 20
E. hshcoltam ..................................................... 21
F. hshrowtam ..................................................... 21
1.7 Real vector and matrix — Elimination .............................. 22
A. elmvec ........................................................ 22
B. elmcol ........................................................ 22
C. elmrow ........................................................ 23
D. elmveccol ..................................................... 23
E. elmcolvec ..................................................... 23
F. elmvecrow ..................................................... 24
G. elmrowvec ..................................................... 24
H. elmcolrow ..................................................... 25
I. elmrowcol ..................................................... 25
J. maxelmrow ..................................................... 26
1.8 Real vector and matrix—Interchanging .............................. 26
A. ichvec ........................................................ 26
B. ichcol ........................................................ 27
C. ichrow ........................................................ 27
D. ichrowcol ..................................................... 28
E. ichseqvec ..................................................... 29
F. ichseq ........................................................ 29
1.9 Real vector and matrix — Rotation ................................. 30
A. rotcol ........................................................ 30
B. rotrow ........................................................ 30
1.10 Real vector and matrix — Norms ................................... 31
A. infnrmvec ..................................................... 31
B. infnrmrow ..................................................... 32
C. infhrmcol ..................................................... 32
D. infnrmmat ..................................................... 33
E. onenrmvec ..................................................... 34
F. onenrmrow ..................................................... 34
G. onenrmcol ..................................................... 35
H. onenrmmat ..................................................... 36
I. absmaxmat ..................................................... 36
1.11 Real vector and matrix — Scaling ................................. 37
reascl ............................................................. 37
1.12 Complex vector and matrix — Multiplication ....................... 38
A. comcolcst ..................................................... 38
B. comrowcst ..................................................... 39
1.13 Complex vector and matrix — Scalar products ...................... 40
A. commatvec ..................................................... 40
B. hshcomcol ..................................................... 40
C. hshcomprd ..................................................... 42
1.14 Complex vector and matrix — Elimination .......................... 43
A. elmcomveccol .................................................. 43
B. elmcomcol ..................................................... 44
C. elmcomrowvec .................................................. 44
1.15 Complex vector and matrix — Rotation ............................. 45
A. rotcomcol ..................................................... 45
B. rotcomrow ..................................................... 46
C. chsh2 ......................................................... 47
1.16 Complex vector and matrix—Norms .................................. 48
comeucnrm .......................................................... 48
1.17 Complex vector and matrix — Scaling .............................. 48
A. comscl ........................................................ 48
B. sclcom ........................................................ 50
1.18 Complex monadic operations ....................................... 51
A. comabs ........................................................ 51
B. comsqrt ....................................................... 51
C. carpol ........................................................ 53
1.19 Complex dyadic operations ........................................ 53
A. commul ........................................................ 53
B. comdiv ........................................................ 54
1.20 Long integer arithmetic .......................................... 55
A. lngintadd ..................................................... 55
B. lngintsubtract ................................................ 56
C. lngintmult .................................................... 58
D. lngintdivide .................................................. 59
E. lngintpower ................................................... 62
2. ALGEBRAIC EVALUATIONS ........................ 64
2.1 Evaluation of polynomials in Grunert form ......................... 64
A. pol ........................................................... 64
B. taypol ........................................................ 65
C. norderpol ..................................................... 66
D. derpol ........................................................ 66
2.2 Evaluation of general orthogonal polynomials ...................... 67
A. ortpol ........................................................ 67
B. ortpolsym ..................................................... 68
C. allortpol ..................................................... 69
D. allortpolsym .................................................. 70
E. sumortpol ..................................................... 70
F. sumortpolsym .................................................. 71
2.3 Evaluation of Chebyshev polynomials ............................... 72
A. chepolsum ..................................................... 72
B. oddchepolsum .................................................. 73
C. chepol ........................................................ 74
D. allchepol ..................................................... 74
2.4 Evaluation of Fourier series ...................................... 75
A. sinser ........................................................ 75
B. cosser ........................................................ 77
C. fouser ........................................................ 78
D. fouser1 ....................................................... 79
E. fbuser2 ....................................................... 80
F. comfouser ..................................................... 80
G. comfouser1 .................................................... 82
H. comfouser2 .................................................... 83
2.5 Evaluation of continued fractions ................................. 84
jfrac .............................................................. 84
2.6 Transformation of polynomial representation ....................... 85
A. polchs ........................................................ 85
B. chspol ........................................................ 86
C. polshtchs ..................................................... 86
D. shtchspol ..................................................... 87
E. grnnew ........................................................ 87
F. newgrn ........................................................ 88
G. lintfmpol ..................................................... 89
2.7 Operations on orthogonal polynomials .............................. 90
intchs ............................................................. 90
3. LINEAR ALGEBRA ............................... 92
3.1 Full real general matrices ........................................ 92
3.1.1 Preparatory procedures ....................................... 92
A. dec ........................................................... 92
B. gsselm ........................................................ 94
C. onenrminv ..................................................... 97
D. erbelm ........................................................ 98
E. gsserb ........................................................ 99
F. gssnri ....................................................... 100
3.1.2 Calculation of determinant .................................. 101
determ .......................................................... 101
3.1.3 Solution of linear equations ................................ 101
A. sol .......................................................... 101
B. decsol ....................................................... 102
C. solelm ....................................................... 103
D. gsssol ....................................................... 104
E. gsssolerb .................................................... 105
3.1.4 Matrix inversion ............................................ 106
A. inv .......................................................... 106
B. decinv ....................................................... 107
C. inv1 ......................................................... 108
D. gssinv ....................................................... 109
E. gssinverb .................................................... 109
3.1.5 Iteratively improved solution ............................... 111
A. itisol ....................................................... 111
B. gssitisol .................................................... 113
C. itisolerb .................................................... 114
D. gssitisolerb ................................................. 116
3.2 Real symmetric positive definite matrices ........................ 118
3.2.1 Preparatory procedures ...................................... 118
A. chldec2 ...................................................... 118
B. chldec1 ...................................................... 119
3.2.2 Calculation of determinant .................................. 120
A. chldeterm2 ................................................... 120
B. chldeterm1 ................................................... 121
3.2.3 Solution of linear equations ................................ 121
A. chlsol2 ...................................................... 121
B. chlsol1 ...................................................... 122
C. chldecsol2 ................................................... 123
D. chldecsol1 ................................................... 124
3.2.4 Matrix inversion ............................................ 125
A. chlinv2 ...................................................... 125
B. chlinv1 ...................................................... 125
C. chldecinv2 ................................................... 126
D. chldecinv1 ................................................... 127
3.3 General real symmetric matrices .................................. 128
3.3.1 Preparatory procedure ....................................... 128
decsym2 ......................................................... 128
3.3.2 Calculation of determinant .................................. 133
determsym2 ...................................................... 133
3.3.3 Solution of linear equations ................................ 133
A. solsym2 ...................................................... 133
B. decsolsym2 ................................................... 135
3.4 Real full rank overdetermined systems ............................ 136
3.4.1 Preparatory procedures ...................................... 136
A. Isqortdec .................................................... 136
B. Isqdglinv .................................................... 138
3.4.2 Least squares solution ...................................... 139
A. Isqsol ....................................................... 139
B. Isqortdecsol ................................................. 140
3.4.3 Inverse matrix of normal equations .......................... 141
Isqinv .......................................................... 141
3.4.4 Least squares with linear constraints ....................... 142
A. Isqdecomp .................................................... 142
B. Isqrefsol .................................................... 146
3.5 Other real matrix problems ....................................... 149
3.5.1 Solution of overdetermined systems .......................... 149
A. solsvdovr .................................................... 149
B. solovr ....................................................... 151
3.5.2 Solution of underdetermined systems ......................... 152
A. solsvdund .................................................... 152
B. solund ....................................................... 153
3.5.3 Solution of homogeneous equation ............................ 154
A. homsolsvd .................................................... 154
B. homsol ....................................................... 155
3.5.4 Pseudo-inversion ............................................ 156
A. psdinvsvd .................................................... 156
B. psdinv ....................................................... 157
3.6 Real sparse nonsymmetric band matrices ........................... 158
3.6.1 Preparatory procedure ....................................... 158
decbnd .......................................................... 158
3.6.2 Calculation of determinant .................................. 161
determbnd ....................................................... 161
3.6.3 Solution of linear equations ................................ 162
A. solbnd ....................................................... 162
B. decsolbnd .................................................... 163
3.7 Real sparse nonsymmetric tridiagonal matrices .................... 166
3.7.1 Preparatory procedures ...................................... 166
A. dectri ....................................................... 166
B. dectripiv .................................................... 167
3.7.2 Solution of linear equations ................................ 170
A. soltri ....................................................... 170
B. decsoltri .................................................... 171
C. soltripiv .................................................... 172
D. decsoltripiv ................................................. 173
3.8 Sparse symmetric positive definite band matrices ................. 176
3.8.1 Preparatory procedure ....................................... 176
chldecbnd ....................................................... 176
3.8.2 Calculation of determinant .................................. 178
chldetermbnd .................................................... 178
3.8.3 Solution of linear equations ................................ 179
A. chlsolbnd .................................................... 179
B. chldecsolbnd ................................................. 180
3.9 Symmetric positive definite tridiagonal matrices ................. 180
3.9.1 Preparatory procedure ....................................... 180
decsymtri ....................................................... 180
3.9.2 Solution of linear equations ................................ 182
A. solsymtri .................................................... 182
B. decsolsymtri ................................................. 183
3.10 Sparse real matrices — Iterative methods ........................ 184
conjgrad ........................................................ 184
3.11 Similarity transformation ....................................... 186
3.11.1 Equilibration - real matrices .............................. 186
A. eqilbr ....................................................... 186
B. baklbr ....................................................... 188
3.11.2 Equilibration - complex matrices ........................... 189
A. eqilbrcom .................................................... 189
B. baklbrcom .................................................... 192
3.11.3 To Hessenberg form - real symmetric ........................ 192
A. tfmsymtri2 ................................................... 192
B. baksymtri2 ................................................... 194
C. tfmprevec .................................................... 195
D. tfmsymtri1 ................................................... 196
E. baksymtri1 ................................................... 198
3.11.4 To Hessenberg form - real asymmetric ....................... 199
A. tfmreahes .................................................... 199
B. bakreahes1 ................................................... 200
C. bakreahes2 ................................................... 201
3.11.5 To Hessenberg form- complex Hermitian ...................... 202
A. hshhrmtri .................................................... 202
B. hshhrmtrival ................................................. 205
C. bakhrmtri .................................................... 207
3.11.6 To Hessenberg form — Complex non-Hermitian ................. 208
A. hshcomhes .................................................... 208
B. bakcomhes .................................................... 210
3.12 Other transformations ........................................... 212
3.12.1 To bidiagonal form - real matrices ......................... 212
A. hshreabid .................................................... 212
B. psttfmmat .................................................... 214
C. pretfmmat .................................................... 214
3.13 The (ordinary) eigenvalue problem ............................... 215
3.13.1 Real symmetric tridiagonal matrices ........................ 215
A. valsymtri .................................................... 215
B. vecsymtri .................................................... 219
C. qrivalsymtri ................................................. 223
D. qrisymtri .................................................... 225
3.13.2 Real symmetric full matrices ............................... 228
A. eigvalsym2 ................................................... 228
B. eigsym2 ...................................................... 229
C. eigvalsym1 ................................................... 230
D. eigsym1 ...................................................... 231
E. qrivalsym2 ................................................... 233
F. qrisym ....................................................... 234
G. qrivalsym1 ................................................... 234
3.13.3 Symmetric matrices - Auxiliary procedures .................. 235
A. mergesort .................................................... 235
B. vecperm ...................................................... 237
C. rowperm ...................................................... 238
3.13.4 Symmetric matrices - Orthogonalization ..................... 239
orthog .......................................................... 239
3.13.5 Symmetric matrices - Iterative improvement ................. 240
symeigimp ....................................................... 240
3.13.6 Asymmetric matrices in Hessenberg form ..................... 244
A. reavalqri .................................................... 244
B. reaveches .................................................... 246
C. reaqri ....................................................... 249
D. comvalqri .................................................... 252
E. comveches .................................................... 255
3.13.7 Real asymmetric full matrices .............................. 258
A. reaeigval .................................................... 258
B. reaeig1 ...................................................... 259
C. reaeig3 ...................................................... 262
D. comeigval .................................................... 263
E. comeig1 ...................................................... 264
3.13.8 Complex Hermitian matrices ................................. 267
A. eigvalhrm .................................................... 267
B. eighrm ....................................................... 268
C. qrivalhrm .................................................... 270
D. qrihrm ....................................................... 271
3.13.9 Complex upper-Hessenberg matrices .......................... 272
A. valqricom .................................................... 272
B. qricom ....................................................... 275
3.13.10 Complex full matrices ..................................... 280
A. eigvalcom .................................................... 280
B. eigcom ....................................................... 281
3.14 The generalized eigenvalue problem .............................. 283
3.14.1 Real asymmetric matrices ................................... 283
A. qzival ....................................................... 283
B. qzi .......................................................... 289
C. hshdecmul .................................................... 298
D. hestgl3 ...................................................... 299
E. hestgl2 ...................................................... 300
F. hsh2col ...................................................... 301
G. hsh3col ...................................................... 302
H. hsh2row3 ..................................................... 304
I. hsh2row2 ..................................................... 305
J. hsh3row3 ..................................................... 307
K. hsh3row2 ..................................................... 308
3.15 Singular values ................................................. 310
3.15.1 Real bidiagonal matrices ................................... 310
A. qrisngvalbid ................................................. 310
B. qrisngvaldecbid .............................................. 312
3.15.2 Real full matrices ......................................... 315
A. qrisngval .................................................... 315
B. qrisngvaldec ................................................. 316
3.16 Zeros of polynomials ............................................ 318
3.16.1 Zeros of general real polynomials .......................... 318
A. zerpol ....................................................... 318
B. bounds ....................................................... 325
3.16.2 Zeros of orthogonal polynomials ............................ 331
A. allzerortpol ................................................. 331
B. lupzerortpol ................................................. 332
C. selzerortpol ................................................. 335
D. alljaczer .................................................... 336
E. alllagzer .................................................... 338
3.16.3 Zeros of complex polynomials ............................... 339
comkwd .......................................................... 339
4. ANALYTIC EVALUATIONS ........................ 341
4.1 Evaluation of an infinite series ................................. 341
A. euler ........................................................ 341
B. sumposseries ................................................. 342
4.2 Quadrature ....................................................... 349
4.2.1 One-dimensional quadrature .................................. 349
A. qadrat ....................................................... 349
B. integral ..................................................... 352
4.2.2 Multidimensional quadrature ................................. 357
tricub .......................................................... 357
4.2.3 Gaussian quadrature - General weights ....................... 361
A. reccof ....................................................... 361
B. gsswts ....................................................... 363
C. gsswtssym .................................................... 364
4.2.4 Gaussian quadrature - Special weights ....................... 366
A. gssjacwghts .................................................. 366
B. gsslagwghts .................................................. 368
4.3 Numerical differentiation ........................................ 369
4.3.1 Calculation with difference formulas ........................ 369
A. jacobnnf ..................................................... 369
B. jacobnmf ..................................................... 371
C. jacobnbndf ................................................... 372
5. ANALYTIC PROBLEMS ........................... 374
5.1 Nonlinear equations .............................................. 374
5.1.1 Single equation - No derivative available ................... 374
A. zeroin ....................................................... 374
B. zeroinrat .................................................... 377
5.1.2 Single equation - Derivative available ...................... 380
zeroinder ....................................................... 380
5.1.3 System of equations - No Jacobian matrix .................... 383
A. quanewbnd .................................................... 383
B. quanewbnd1 ................................................... 387
5.2 Unconstrained optimization ....................................... 389
5.2.1 One variable -No derivative ................................. 389
minin ........................................................... 389
5.2.2 One variable - Derivative available ......................... 393
mininder ........................................................ 393
5.2.3 More variables - Auxiliary procedures ....................... 396
A. linemin ...................................................... 396
B. rnk1upd ...................................................... 399
C. davupd ....................................................... 400
D. fleupd ....................................................... 401
5.2.4 More variables - No derivatives ............................. 402
praxis .......................................................... 402
5.2.5 More variables - Gradient available ......................... 411
A. rnk1min ...................................................... 411
B. flemin ....................................................... 418
5.3 Overdetermined nonlinear systems ................................. 422
5.3.1 Least squares - With Jacobian matrix ........................ 422
A. marquardt .................................................... 422
B. gssnewton .................................................... 427
5.4 Differential equations — Initial value problems .................. 433
5.4.1 First order - No derivatives right hand side ................ 433
A. rk1 .......................................................... 433
B. rke .......................................................... 436
C. rk4a ......................................................... 440
D. rk4na ........................................................ 448
E. rk5na ........................................................ 455
F. multistep .................................................... 461
G. diffsys ...................................................... 473
H. ark .......................................................... 477
I. efrk ......................................................... 483
5.4.2 First order - Jacobian matrix available ..................... 492
A. efsirk ....................................................... 492
B. eferk ........................................................ 497
C. liniger1vs ................................................... 504
D. Iiniger2 ..................................................... 511
E. gms .......................................................... 517
F. impex ........................................................ 526
5.4.3 First order - Several derivatives available ................. 539
A. modifiedtaylor ............................................... 539
B. eft .......................................................... 545
5.4.4 Second order - No derivatives right hand side ............... 555
A. rk2 .......................................................... 555
B. rk2n ......................................................... 559
C. rk3 .......................................................... 564
D. rk3n ......................................................... 567
5.4.5 Initial boundary value problem .............................. 572
arkmat .......................................................... 572
5.5 Two point boundary value problems ................................ 576
5.5.1 Linear methods - Second order self adjoint ................... 576
A. femlagsym .................................................... 576
B. femlag ....................................................... 584
C. femlagspher .................................................. 589
5.5.2 Linear methods - Second order skew adjoint .................. 595
femlagskew ...................................................... 595
5.5.3 Linear methods - Fourth order self adjoint .................. 601
femhermsym ...................................................... 601
5.5.4 Nonlinear methods ........................................... 612
nonlinfemlagskew ................................................ 612
5.6 Two-dimensional boundary value problems .......................... 617
5.6.1 Elliptic special linear systems ............................. 617
A. richardson ................................................... 617
B. elimination .................................................. 621
5.7 Parameter estimation in differential equations ................... 627
5.7.1 Initial value problems ....................................... 627
peide ........................................................... 627
6. SPECIAL FUNCTIONS ........................... 651
6.1 Elementary functions ............................................. 651
6.1.1 Hyperbolic functions ........................................ 651
A. arcsinh ...................................................... 651
B. arccosh ...................................................... 652
C. arctanh ...................................................... 652
6.1.2 Logarithmic functions ....................................... 653
logoneplusx ..................................................... 653
6.2 Exponential integral ............................................. 654
6.2.1 Exponential integral ........................................ 654
A. ei ........................................................... 654
B. eialpha ...................................................... 657
C. enx .......................................................... 657
D. nonexpenx .................................................... 659
6.2.2 Sine and cosine integral .................................... 661
A. sincosint .................................................... 661
B. sincosfg ..................................................... 662
6.3 Gamma function ................................................... 664
A. recipgamma ................................................... 664
B. gamma ........................................................ 665
C. loggamma ..................................................... 667
D. incomgam ..................................................... 668
E. incbeta ...................................................... 671
F. ibpplusn ..................................................... 672
G. ibqplusn ..................................................... 673
H. ixqfix ....................................................... 674
I. ixpfix ....................................................... 675
J. forward ...................................................... 676
K. backward ..................................................... 676
6.4 Error function ................................................... 677
A. errorfunction ................................................ 677
B. nonexperfc ................................................... 678
C. inverseerrorfunction ......................................... 680
D. fresnel ...................................................... 682
E. fg ........................................................... 684
6.5 Bessel functions of integer order ................................ 686
6.5.1 Bessel functions J and Y .................................... 686
A. bessj0 ....................................................... 686
B. bessj1 ....................................................... 688
C. bessj ........................................................ 689
D. bessy01 ...................................................... 691
E. bessy ........................................................ 692
F. besspq0 ...................................................... 693
G. besspql ...................................................... 695
6.5.2 Bessel functions I and K .................................... 696
A. bessi0 ....................................................... 696
B. bessi1 ....................................................... 697
C. bessi ........................................................ 698
D. bessk01 ...................................................... 699
E. bessk ........................................................ 700
F. nonexpbessi0 ................................................. 701
G. nonexpbessi1 ................................................. 702
H. nonexpbessi .................................................. 704
I. nonexpbessk01 ................................................ 705
J. nonexpbessk .................................................. 707
6.6 Bessel functions of real order ................................... 708
6.6.1 Bessel functions J and Y .................................... 708
A. bessjaplusn .................................................. 708
B. bessya01 ..................................................... 709
C. bessyaplusn .................................................. 712
D. besspqa01 .................................................... 713
E. besszeros .................................................... 716
F. start ........................................................ 719
6.6.2 Bessel functions I and K .................................... 721
A. bessiaplusn .................................................. 721
B. besska01 ..................................................... 722
C. besskaplusn .................................................. 724
D. nonexpbessiaplusn ............................................ 725
E. nonexpbesska01 ............................................... 726
F. nonexpbesskaplusn ............................................ 728
6.6.3 Spherical Bessel functions .................................. 729
A. spherbessj ................................................... 729
B. spherbessy ................................................... 730
C. spherbessi ................................................... 731
D. spherbessk ................................................... 731
E. nonexpspherbessi ............................................. 732
F. nonexpspherbessk ............................................. 733
6.6.4 Airy functions .............................................. 734
A. airy ......................................................... 734
B. airyzeros .................................................... 738
7. INTERPOLATION AND APPROXIMATION ............. 741
7.1 Real data in one dimension ....................................... 741
7.1.1 Interpolation with general polynomials ...................... 741
newton .......................................................... 741
7.1.2 Approximation in infinity norm .............................. 742
A. ini .......................................................... 742
B. sndremez ..................................................... 743
C. minmaxpol .................................................... 746
ADDENDA ........................................ 749
I. Fast Fourier transforms ........................................... 749
A. cfftp .......................................................... 749
B. orderf ......................................................... 762
C. cfft2p ......................................................... 764
D. cfft2r ......................................................... 767
E. Test_cfftp ..................................................... 772
F. rfftr .......................................................... 776
G. Test_rfftr ..................................................... 778
II. Time series analysis ............................................. 780
A. powsp .......................................................... 780
B. Test_powsp ..................................................... 784
C. timser ......................................................... 786
D. Test_timser .................................................... 789
E. timspc ......................................................... 790
F. Test_timspc .................................................... 799
WORKED EXAMPLES ................................ 804
Examples for chapter 1 procedures .................................... 804
hshcomcol, hshcomprd .............................................. 804
elmcomcol ......................................................... 805
rotcomcol ......................................................... 806
comabs ............................................................ 807
comsqrt ........................................................... 807
carpol ............................................................ 808
commul ............................................................ 808
comdiv ............................................................ 809
Ingintadd, Ingintsbtract, Ingintmult, lngintdivide, lngintpower ... 810
Examples for chapter 2 procedures .................................... 811
derpol ............................................................ 811
allortpol ......................................................... 812
chepolsum ......................................................... 812
oddchepolsum ...................................................... 813
chepol, allchepol ................................................. 814
fouser ............................................................ 815
jfrac ............................................................. 815
chspol, polchs .................................................... 816
polshtchs, shtchspol .............................................. 817
newgrn, grnnew .................................................... 818
lintfmpol ......................................................... 818
intchs ............................................................ 819
Examples for chapter 3 procedures .................................... 820
determ, gsselm .................................................... 820
decsol ............................................................ 821
gsssol ............................................................ 822
gsssolerb ......................................................... 823
decinv ............................................................ 824
gssinv ............................................................ 825
gssinverb ......................................................... 826
gssitisol ......................................................... 827
gssitisolerb ...................................................... 828
chldec2, chlsol2, chlinv2 ......................................... 829
chldec1, chlsol1, chlinv1 ......................................... 831
chldecsol2, chldeterm2, chldecinv2 ............................... 832
chldecsol1, chldeterm1, chldecinv1 ................................ 833
determsym2 ........................................................ 835
decsolsym2 ........................................................ 836
lsqortdec, lsqsol, lsqdglinv ...................................... 838
lsqortdecsol ...................................................... 839
lsqinv ............................................................ 841
lsqdecomp, lsqrefsol .............................................. 842
solovr ............................................................ 844
solund ............................................................ 845
homsol ............................................................ 847
psdinv ............................................................ 849
solbnd, decbnd, determbnd ......................................... 850
decsolbnd ......................................................... 851
decsoltri ......................................................... 852
soltripiv ......................................................... 853
decsoltripiv ...................................................... 854
chlsolbnd, chldecbnd, chldetermbnd ................................ 855
chldecsolbnd, chldetermbnd ........................................ 856
decsolsymtri ...................................................... 857
conjgrad .......................................................... 858
eqilbrcom ......................................................... 860
hshhrmtri ......................................................... 861
valsymtri, vecsymtri .............................................. 862
eigsyml ........................................................... 863
symeigimp ......................................................... 865
comvalqri, comveches .............................................. 866
reaeig3 ........................................................... 868
eighrm ............................................................ 869
qrihrm ............................................................ 871
valqricom ......................................................... 872
qricom ............................................................ 874
eigcom ............................................................ 875
qzival ............................................................ 877
qzi ............................................................... 878
qrisngvaldec ...................................................... 880
zerpol, bounds .................................................... 881
allzerortpol ...................................................... 883
lupzerortpol ...................................................... 884
selzerortpol ...................................................... 885
alljaczer ......................................................... 886
alllagzer ......................................................... 887
comkwd ............................................................ 887
Examples for chapter 4 procedures .................................... 888
euler ............................................................. 888
sumposseries ...................................................... 889
qadrat ............................................................ 890
integral .......................................................... 890
tricub ............................................................ 892
reccof ............................................................ 893
gsswtssym ......................................................... 894
gssjacwghts ....................................................... 895
gsslagwghts ....................................................... 895
jacobnnf .......................................................... 896
jacobnmf .......................................................... 897
jacobnbndf ........................................................ 899
Examples for chapter 5 procedures .................................... 900
zeroin ............................................................ 900
zeroinrat ......................................................... 901
zeroinder ......................................................... 902
quanewbnd1 ........................................................ 903
minin ............................................................. 904
mininder .......................................................... 906
praxis ............................................................ 907
rnk1min, flemin ................................................... 908
marquardt ......................................................... 910
gssnewton ......................................................... 912
rkl ............................................................... 914
rke ............................................................... 915
rk4a .............................................................. 917
rk4na ............................................................. 918
rk5na ............................................................. 920
multistep ......................................................... 921
diffsys ........................................................... 923
ark ............................................................... 925
efrk .............................................................. 928
eisirk ............................................................ 930
eferk ............................................................. 932
liniger1vs ........................................................ 934
liniger2 .......................................................... 936
gms ............................................................... 940
impex ............................................................. 942
modifiedtaylor .................................................... 945
eft ............................................................... 948
rk2 ............................................................... 952
rk2n .............................................................. 953
rk3 ............................................................... 955
rk3n .............................................................. 956
arkmat ............................................................ 958
femlagsym ......................................................... 961
femlag ............................................................ 963
femlagspher ....................................................... 964
femlagskew ........................................................ 966
femhermsym ........................................................ 968
nonlinfemlagskew .................................................. 970
richardson ........................................................ 972
elimination ....................................................... 974
peide ............................................................. 976
Examples for chapter 6 procedures ................................... 984
ei ................................................................ 984
eialpha ........................................................... 984
enx, nonexpenx .................................................... 985
sincosint, sincosfg ............................................... 986
recipgamma ........................................................ 987
gamma ............................................................. 988
loggamma .......................................................... 989
incomgam .......................................................... 989
incbeta ........................................................... 990
ibpplusn .......................................................... 991
ibqplusn .......................................................... 991
errorfunction, nonexperfc ......................................... 992
inverseerrorfunction .............................................. 993
fresnel, fg ....................................................... 994
bessj0, bessj1, bessj ............................................. 995
bessy01 ........................................................... 995
bessy ............................................................. 996
besspq0, besspq1 .................................................. 997
bessi, bessk ...................................................... 998
bessk01 ........................................................... 999
nonexpbessk01 .................................................... 1000
nonexpbessk ...................................................... 1001
bessjaplusn ...................................................... 1001
besspqa01 ........................................................ 1002
besszeros ........................................................ 1003
spherbessi, nonexpspherbessi ..................................... 1004
spherbessk, nonexpspherbessk ..................................... 1005
airy ............................................................. 1006
airyzeros ........................................................ 1007
Examples for chapter 7 procedures .................................. 1007
newton ........................................................... 1007
ini .............................................................. 1008
sndremez ......................................................... 1009
minmaxpol ........................................................ 1010
APPENDIX A: REFERENCES ....................... 1012
APPENDIX B: PROCEDURES DESCRIPTION ........... 1021
INDEX OF PROCEDURES .......................... 1055
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