SSL II USER'S GUIDE - Lahey Computer Systems

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$SSL2 Extended Capabilities Math Library User's Guide - Lahey ...$

C. Nonlinear Equations Subroutine name Item Page Remarks RQDR Zeros of a quadratic with real coefficients 552 CQDR Zeros of a quadratic with complex coefficients 286 LOWP Zeros of a low degree polynomial with real coefficients (fifth degree or lower) 423 RJETR Zeros of a polynomial with real coefficients (Jenkins-Traub method) 546 CJART Zeros of a polynomial with complex coefficients (Jarratt method) 275 TSD1 D. Extrema Zero of a real function which changes sign in a given interval (derivative not required) TSDM Zero of a real function (Muller’s method) 601 CTSDM Zero of complex function (Muller’s method) 292 NOLBR Solution of a system of nonlinear equations (Brent’s method) 487 Subroutine name Item Page Remarks LMINF LMING MINF1 MING1 NOLF1 NOLG1 Minimization of function with a variable (quadratic interpolation using function values only) Minimization of function with a variable (cubic interpolation using function values and its derivatives) Minimization of function with several variables (revised quasi-Newton method, uses function values only) Minimization of a function with several variables (Quasi-Newton method, using function values and its derivatives) Minimization of the sum of squares of functions with several variables (Revised Marquardt method, using function values only) Minimization of the sum of squares of functions (revised Marquardt method using function values and its derivatives) LPRS1 Solution of a linear programming problem (Revised simplex method) 425 NLPG1 Nonlinear programming (Powell’s method using function values and its derivatives) 482 E. Interpolation and Approximation Interpolation Subroutine name Item Page Remarks AKLAG Aitken-Lagrange interpolation 89 AKHER Aitkan-Hermite interpolation 86 SPLV Cubic spline interpolation 579 BIF1 B-spline interpolation (I) 156 BIF2 B-spline interpolation (II) 158 BIF3 B-spline interpolation (III) 160 BIF4 B-spline interpolation (IV) 162 BIFD1 B-spline two-dimensional interpolation (I-I) 151 BIFD3 B-spline two-dimensional interpolation (III-III) 154 AKMID Two-dimensional quasi-Hermite Interpolation 91 INSPL Cubic spline interpolation coefficient calculation 377 AKMIN Quasi-Hermite interpolation coefficient calculation 95 BIC1 B-spline interpolation coefficient calculation (I) 143 BIC2 B-spline interpolation coefficient calculation (II) 145 BIC3 B-spline interpolation coefficient calculation (III) 147 BIC4 B-spline interpolation coefficient calculation (IV) 149 BICD1 B-spline two-dimensional interpolation coefficient calculation (I-I) 138 BICD3 B-spline two-dimensional interpolation coefficient calculation (III-III) 141 604 418 420 466 470 490 494 5
Approximation 6 Subroutine name Item Page Remarks LESQ1 Polynomial least squares approximation 416 Smoothing Subroutine name Item Page Remarks SMLE1 Data smoothing by local least squares polynomials (equally spaced data points) 575 SMLE2 Data smoothing by local least squares polynomials (unequally spaced data points) 577 BSF1 B-spline smoothing 216 BSC1 B-spline smoothing coefficient calculation 201 BSC2 B-spline smoothing coefficient calculation (variable knots) 203 BSFD1 B-spline two-dimensional smoothing 214 BSCD2 B-spline two-dimensional smoothing coefficient calculation (variable knots) 194 Series Subroutine name Item Page Remarks FCOSF Fourier Cosine series expansion of an even function (Function input, fast cosine transform) ECOSP Evaluation of a cosine series 296 FSINF Fourier sine series expansion of an odd function (Function input, fast sine transform) 324 ESINP Evaluation of a sine series 302 FCHEB Chabyshev series expansion of a real function (Function input, fast cosine transform) 306 ECHEB Evaluation of a Chebyshev series 294 GCHEB Differentiation of a Chebyshev series 339 ICHEB Indefinite integral of a Chebyshev series 367 312
Page 1 and 2: SSL II USER’S GUIDE (Scientific S
Page 3 and 4: This document contains technology r
Page 6 and 7: CONTENTS SSL II SUBROUTINE LIST....
Page 8: 11.3 Exponential Integral .........
Page 11 and 12: Linear equations 2 Subroutine name
Page 13: Eigenvalues 4 Subroutine name Item
Page 17 and 18: Numerical Quadrature 8 Subroutine n
Page 19: J. Pseudo Random Numbers 10 Subrout
Page 22 and 23: (as for the definition of them, see
Page 24: PART I GENERAL DESCRIPTION
Page 27 and 28: GENERAL DESCRIPTION eigenvector cal
Page 29 and 30: GENERAL DESCRIPTION 20 This alphanu
Page 31 and 32: GENERAL DESCRIPTION Matrix definiti
Page 33 and 34: GENERAL DESCRIPTION as shown below,
Page 35 and 36: GENERAL DESCRIPTION ⎡ x1 ⎤ ⎢
Page 37 and 38: CHAPTER 3 LINEAR ALGEBRA 3.1 OUTLIN
Page 39 and 40: GENERAL DESCRIPTION • Determinant
Page 41 and 42: GENERAL DESCRIPTION • Transpositi
Page 43 and 44: GENERAL DESCRIPTION 34 rank (A) = r
Page 45 and 46: GENERAL DESCRIPTION In addition, ob
Page 47 and 48: GENERAL DESCRIPTION 38 routine, how
Page 49 and 50: GENERAL DESCRIPTION 40 implicitly.
Page 51 and 52: GENERAL DESCRIPTION where h is the
Page 53 and 54: GENERAL DESCRIPTION 44 Table 4.7 Su
Page 55 and 56: CHAPTER 5 NONLINEAR EQUATIONS 5.1 O
Page 57 and 58: GENERAL DESCRIPTION 48 Table 5.3 No
Page 59 and 60: GENERAL DESCRIPTION 50 * 1 * T * f
Page 61 and 62: GENERAL DESCRIPTION Comments on use
Page 63 and 64: GENERAL DESCRIPTION 54 (that is, at
Page 65 and 66:
GENERAL DESCRIPTION 56 is defined a
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GENERAL DESCRIPTION 58 − Calculat
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GENERAL DESCRIPTION 60 y y ( λ,
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GENERAL DESCRIPTION 7.5 SERIES SSL
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GENERAL DESCRIPTION 64 which defini
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GENERAL DESCRIPTION 8.6 LAPLACE TRA
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GENERAL DESCRIPTION Since G(s) is r
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GENERAL DESCRIPTION CHAPTER 9 NUMER
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GENERAL DESCRIPTION Table 9.2 Numer
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GENERAL DESCRIPTION and those combi
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GENERAL DESCRIPTION • Final value
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CHAPTER 11 SPECIAL FUNCTIONS 11.1 O
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GENERAL DESCRIPTION 11.8 BESSEL FUN
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GENERAL DESCRIPTION Table 12.2 Pseu
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A21-11-0101 AGGM, DAGGM Addition of
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value of Zl, l, |Dl| and − if |Dj
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E11-11-0101 AKLAG, DAKLAG Aitken-La
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E11-42-0101 AKMID, DAKMID Two-dimen
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p q p q f 0 1 1 2 () t = 1 − 3t 2
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E12-21-0201 AKMIN, DAKMIN Quasi-Her
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y−1 y0 y1 x−1 x0 x1 x2 x3 Fig.
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C **EXAMPLE** DIMENSION A(100,100),
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This purpose is sometimes obstructe
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By regarding this as a system of li
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Therefore, if the following holds,
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NMIN ≤ Number of evaluations ≤
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holds takes the form. Thus, if the
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example NMAX=2, NMAX is automatical
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Comments on use • Subprograms use
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G24-13-0101 AQMC8, DAQMC8 Multiple
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CALL AQMC8(M,LSUB,FUN,EPSA,EPSR, *N
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I = α + α 1 1∫−1 ∫ 1 1 −1
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G24-13-0201 AQME, DAQME Multiple in
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• Notes The function subprogram a
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• Computing procedure Procedure 1
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NMIN≤Number of evaluations≤NMAX
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much more accurate than required. I
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A21-12-0101 ASSM, DASSM Addition of
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perform singular value decompositio
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The first rotating angle φ 2, that
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500 FORMAT(2I5) 510 FORMAT(4E15.7)
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Table BICD1-1 Condition codes Code
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E12-32-3302 BICD3, DBICD3 B-spline
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E12-31-0102 BIC1, DBIC1 B-spline in
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Page 156 and 157:
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E11-32-1101 BIFD1, DBIFD1 B-spline
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WRITE(6,650) IXX,IYY,((R(I,J),I=1,5
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Table BIFD3-1 Condition codes Code
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DO 40 L2=1,M2 ISW=L2-2 WRITE(6,620)
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DO 30 I=1,N1 H=(X(I+1)-X(I))/5.0 XI
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DO 20 J=1,6 V=XI+H*FLOAT(J-1) II=I
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DO 30 I=1,N1 H=(X(I+1)-X(I))/5.0 XI
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The value of In(x) is computed as a
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I11-81-0601 BI0,DBI0 Zero order mod
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I11-81-1001 BJN, DBJN Integer order
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I11-83-0101 BJR, DBJR Real order Be
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I11-81-0201 BJ0,DBJ0 Zero order Bes
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I11-81-0301 BJ1, DBJ1 First order B
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I11-81-1301 BKN, DBKN Integer order
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I v () x ⎧ v ⎪ ⎛ x ⎞ ⎪ 1
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Equation (4.29) can be expressed us
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I11-81-0901 BK1, DBK1 First order m
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=(aij (s) ), balancing is performed
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for the constant vector). However,
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A52-11-0202 BLU1, DBLU1 LU-decompos
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where ⎡1 ⎤ ⎢ ⎥ ⎢ 1 0 ⎥
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• Example Simultaneous linear equ
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where ( x y )( ) s1= n + n + 2m m+
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Replacing the updated knots in the
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For the matrix (T−λ I) shown in
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E32-31-0102 BSC1, DBSC1 B-spline sm
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E32-31-0202 BSC2, DBSC2 B-spline sm
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may assume the minimum. (For detail
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C **EXAMPLE** DIMENSION A(1100),E(1
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attained. If so, it is safe to choo
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Z T Z=P T C T ΛΛΛΛ 2 CP=LL T Wh
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Jennings' acceleration is effective
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obtained by subroutine BSCD2. There
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DO 10 I=1,N 10 W(I)=1.0 CALL BSC1 (
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Given µj as an approximation of λ
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B51-21-0302 BTRID, DBTRID Reduction
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I11-81-1101 BYN, DBYN Integer order
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J ⎛ 1 ⎜− + ⎝ 4 2! Γ 3 −
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a 0 l = 1 l a = i ⎪ 2 2 2 2 2 2 (
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Theoretical precision = 10.66 digit
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Q 1 1 = ∑ k 2 ∑ k= 0 k= 0 2k +
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I11-82-1301 CBJN, DCBJN Integer ord
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is repeatedly applied to k=m,m-1,..
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• For: Single precision: |Im(z)|
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B21-15-0202 CBLNC, DCBLNC Balancing
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I11-82-1401 CBYN, DCBYN Integer ord
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IF(MM.EQ.0) GO TO 50 DO 40 INT=1,MM
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I11-11-0201 CELI2, DCELI2 Complete
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F12-15-0101 CFT, DCFT Multi-variate
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In this routine, the 8 and 2 radix
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where definitions of K1, ..., KM, J
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= n 2 2 = r ⋅ q + q ⋅ r + ( q
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without permutation. So, by the del
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Method This subroutine performs dis
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F12-15-0302 CFTR, DCFTR Discrete co
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NT ( = N1× N2× N3) * One-dimensio
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A11-40-0101 CGSBM, DCGSBM Storage m
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Elements in the general mode Elemen
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DO 30 J=1,N IM=MIN0(J+1,N) DO 30 I=
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Method An n-order complex matrix A
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The process for the complex QR meth
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C **EXAMPLE** COMPLEX ZA(100,100),Z
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C22-15-0101 CJART, DCJART Zeros of
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A22-15-0202 CLU, DCLU LU-decomposit
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A22-15-0602 CLUIV, DCLUIV The inver
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A22-15-0302 CLUX, DCLUX A system of
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B21-15-0702 CNRML, DCNRML Normaliza
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I11-41-0201 COSI, DCOSI Cosine inte
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A11-40-0201 CSBGM, DCSBGM Storage m
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A11-50-0201 CSBSM, DCSBSM Storage m
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A11-50-0101 CSSBM, DCSSBM Storage c
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Method The subroutine uses Muller
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FUNCTION FUN(X) REAL*8 SUM,XP,TERM
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WRITE(6,611) X,P,Q 3 CONTINUE W=W+W
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Therefore if real part uj and imagi
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From matrix F thus obtained and tra
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601 FORMAT(/(5E15.5)) 610 FORMAT('0
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Theoretical precision = 10.04 digit
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trigonometric function values neces
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in the sense that it can be utilize
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Step 4: Discrete cosine transform (
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the maximum value of (power of 2) +
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Method This subroutine applies disc
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It also calculates norm f based on
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• Example By inputting n sample p
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F11-11-0101 FCOST, DFCOST Discrete
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The equations shown in (4.3) can be
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The degree of error decerement grea
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Method This subroutine applies disc
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terms is about nlog2 n. To save sto
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Even when the dimension differs, th
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F11-21-0101, FSINT, DFSINT Discrete
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B52-11-0101 GBSEG, DGBSEG Eigenvalu
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For detailed information about the
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E51-30-0301 GCHEB, DGCHEB Different
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A25-31-0101 GINV, DGINV Moore-Penro
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B22-10-0402 GSBK, DGSBK Back transf
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B22-21-0302 GSCHL, DGSCHL Reduction
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B22-21-0201 GSEG2, DGSEG2 Eigenvalu
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The above processing are accomplish
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Comments • Subprogram used SSL II
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In this case, the current stepsize
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For further information, refer to t
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WRITE(6,630) (J,J=INT,LST) WRITE(6,
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Method An n-order real matrix A is
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B21-11-0402 HSQR,DHSOR Eigenvalues
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B21-11-0502 HVEC,DHVEC Eigenvectors
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CALL HVEC(A,100,N,ER,EI,IND,M,EV,10
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E51-30-0401 ICHEB, DICHEB Indefinit
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I11-71-0301 IERF, DIERF Inverse err
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(c) 2.5 10 -3 ≤x
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I11-61-0201 IGAM2, DIGAM2 Incomplet
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I11-91-0301 INDF, DINDF Inverse nor
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E12-21-0101 INSPL, DINSPL Cubic spl
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F20-01-0101 LAPS1, DLAPS1 Inversion
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Page 392 and 393:
Page 394 and 395:
In each of the above, the delay fac
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This subroutine calculates f(t) for
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10 READ(5,510) (B(I),I=1,N) WRITE(6
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610 FORMAT(///10X,'CONSTANT VECTOR=
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A25-21-0101 LAXLM, DLAXLM Least squ
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Where ~ b is an n-dimensional vecto
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A25-11-0401 LAXLR, DLAXLR Iterative
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A22-11-0401 LAXR, DLAXR Iterative r
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esults, this indicates that the con
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• Example In this example, l syst
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Table LBX1R-1 Condition codes code
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A22-15-0101 LCX,DLCX A system of li
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A22-15-0401 LCXR,DLCXR Iterative re
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x (1) is e (1) (=x (1) − x), its
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Method Given the LDL T decomposed m
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Method To solve a system of linear
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500 FORMAT(I5) 510 FORMAT(2F10.0) 6
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C **EXAMPLE** EXTERNAL FUN A=-5.0 B
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e resumed by calling this subroutin
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C21-41-0101 LOWP, DLOWP Zeros of a
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D21-10-0101 LPRS1, DLPRS1 Linear pr
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(which is derived form other constr
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• Basic inveerse matrix B -1 for
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A52-21-0101 LSBIX, DLSBIX A system
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A52-31-0101 LSBX, DLSBX A system of
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A52-31-0401 LSBXR, DLSBXR Iterative
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• Accuracy estimation for approxi
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C **EXAMPLE** DIMENSION A(5050),B(1
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• Example An approximate solution
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This subroutine make use of the cha
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A22-51-0101 LSX, DLSX A system of l
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A22-51-0401 LSXR, DLSXR Iterative r
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A52-11-0501 LTX, DLTX A system of l
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( ) m k+ 1 ,k ( k ) ( k = a ) k+ 1,
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Method This subroutine computes the
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500 FORMAT(I5) 510 FORMAT(4E15.7) 6
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y n i =∑ j= 1 a ij x j , i = 1,..
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The subroutines PBM and PGM are use
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SUBROUTINE PCM(ICOM,L,ZA,K,M,N) DIM
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The subroutines PSM and PGM in this
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A21-11-0401 MGSM, DMGSM Multiplicat
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for the iteration vector xk and if
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• If 2α k < α min < 4α k , the
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Giving MAX: For a variable vector x
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ϕ (α) f 0 f 1 f 2 α k 2α k 4α
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Method The multiplication ( y ) is
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A21-12-0301 MSSM, DMSSM Multiplicat
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Method This subroutine performs mul
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I11-91-0201 NDFC, DNDFC Complementa
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Table NLPG1-1 Condition codes Code
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( x , λλλλ ) ∇ φ( x λλλλ
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C24-11-0101 NOLBR, DNOLBR Solution
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k T ( y) ≈ f ( y ) + g ( y − )
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value for an additional evaluation
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2) To compute the difference approx
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MAX The function evaluation count i
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m n 2 ∑∑( fi / x j ) / ( mn) v
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500 FORMAT(I5) 510 FORMAT(5E15.7) 6
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EPSA ..... Input. Absolute error to
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IF(ICON.EQ.30000) STOP GO TO 10 20
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c ( x − xm )( x − x−1 ) ... (
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− k > 2 and max (ERKM1, ERKM2)
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H11-20-0151 ODGE, DODGE A stiff sys
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Table ODGE-1 Condition codes Code M
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Method Both Gear’s and Adams meth
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xm+1 is determined, solution ye at
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8) Corrector iteration This subrout
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Table ODRK1-1 Condition codes Code
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h′ = + { 6 min hi hi fi ( x0, y0
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Thus, this subroutine is used with
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J12-20-0101 RANB2 Generation of bin
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J11-30-0101 RANE2 Generation of exp
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2) For u − 0 . 5 > 0. 46875 G -1
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J12-10-0101 RANP2 Generation of Poi
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J11-10-0101 RANU2 Generation of uni
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There is a theory that such a perio
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is coupled such as Pi (IXi, IXi+1)
Page 548 and 549:
m = m1 + m2 + ... + ms One call of
Page 550 and 551:
One call of this subroutine enables
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F11-31-0101 RFT, DRFT Discrete real
Page 554 and 555:
from α α α x1 k x2 k x1 0 1 =
Page 556 and 557:
faster than second order. • K pol
Page 558 and 559:
Letting ( ) λ 2 ( ) σ ( x) = x +
Page 560 and 561:
C **EXAMPLE** DIMENSION Y(3,10),F(3
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A52-31-0202 SBDL, DSBDL LDL T -deco
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A52-21-0202 SBMDM, DSBMDM MDM T dec
Page 566 and 567:
1) Determines 2) If ( k − 1) ( k
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DO 10 I=1,N WRITE(6,630) I,(EV(I,J)
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eigenvectors are determined using t
Page 572 and 573:
A21-11-0201 SGGM, DSGGM Subtraction
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G23-11-0101 SIMP2, DSIMP2 Integrati
Page 576 and 577:
a ( 4, 2 ) ( 2, 1 ) ( 5, 2 ) ( 1, 0
Page 578 and 579:
I11-41-0101 SINI, DSINI Sine integr
Page 580 and 581:
C **EXAMPLE** DIMENSION A(5050) 10
Page 582 and 583:
i.e., -s is stored in IP (k+1). The
Page 584 and 585:
E31-11-0101 SMLE1, DSMLE1 Data smoo
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E31-21-0101 SMLE2, DSMLE2 Data smoo
Page 588 and 589:
E11-21-0101 SPLV, DSPLV Cubic splin
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(c) (ISW(1) =3,DY (1)=, f"( x1)/f"(
Page 592 and 593:
B21-21-0602 TEIG1, DTEIG1 Eigenvalu
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B21-21-0702 TEIG2, DTEIG2 Selected
Page 596 and 597:
• Direct sum of submatrices If T
Page 598 and 599:
B21-21-0802 TRBK, DTRBK Back transf
Page 600 and 601:
B21-25-0402 TRBKH, DTRBKH Back tran
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B21-25-0302 TRIDH, DTRIDH Reduction
Page 604 and 605:
where, l = n − k + 1 1 2 1 = k tl
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( ( ) ) ( ( ) ) ( ( ) ) 2 σ k k 2
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T= d 1 e 1 e 1 d 2 e 2 0 e 2 e n−
Page 610 and 611:
C23-11-0111 TSDM, DTSDM Zero of a r
Page 612 and 613:
• Convergence criteria The follow
Page 614 and 615:
Variables a, b, and c have the foll
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APPENDIX A AUXILIARY SUBROUTINES A.
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A.3 MGSSL Printing of condition mes
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A.5 ASUM(BSUM), DSUM(DBSUM) Product
Page 622 and 623:
A.7 IRADIX Radix of the floating-po
Page 624 and 625:
APPENDIX B ALPHABETIC GUIDE FOR SUB
Page 626 and 627:
TableB.1-conteimued Subroutine Clas
Page 628 and 629:
B.3 AUXILIARY SUBROUTINES Table B.3
Page 630 and 631:
Interpolation and approximation Cod
Page 632 and 633:
[20] Brezinski, C. Acceleration de
Page 634 and 635:
[82] Uno,T. Bessel Function. Baifuk
Page 636 and 637:
Author Subroutine Item T. Torii ECH
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