SSL II USER'S GUIDE - Lahey Computer Systems

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

A. Linear Algebra SSL II SUBROUTINE LIST The SSL II functions are listed below. Generally, a single-precision routine and a double-precision routine are available for each function. The subroutine name column gives the names of single-precision routines. Double-precision routine names start with a D, followed by the single-precision names. If the use of a function is restricted due to hardware restrictions, it is indicated in the remarks column. The symbols that appear in the remarks column mean the following: #: Only the single-precision routine is available in all systems. Storage mode conversion of matrices Subroutine name Item Page Remarks CGSM Storage mode conversion of matrices (real symmetric to real general) 264 CSGM Storage mode conversion of matrices (real general to real symmetric) 290 CGSBM Storage mode conversion of matrices (real general to real symmetric band) 263 CSBGM Storage mode conversion of matrices (real symmetric band to real general) 287 CSSBM Storage mode conversion of matrices (real symmetric to real symmetric band) 291 CSBSM Storage mode conversion of matrices (real symmetric band to real symmetric) 289 Matrix manipulation Subroutine name Item Page Remarks AGGM Addition of two matrices (real general + real general) 85 SGGM Subtraction of two matrices (real general – real general) 563 MGGM Multiplication of two matrices (real general by real general) 454 MGSM Multiplication of two matrices (real general by real symmetric) 465 ASSM Addition of two matrices (real symmetric + real symmetric) 131 SSSM Subtraction of two matrices (real symmetric – real symmetric) 582 MSSM Multiplication of two matrices (real symmetric by real symmetric) 477 MSGM Multiplication of two matrices (real symmetric by real general) 476 MAV Multiplication of a real matrix by a real vector 456 MCV Multiplication of a complex matrix by a complex vector 460 MSV Multiplication of a real symmetric matrix by a real vector 478 MSBV Multiplication of a real symmetric band matrix and a real vector 474 MBV Multiplication of a real band matrix and a real vector 458 1
Linear equations 2 Subroutine name Item Page Remarks LAX A system of linear equations with a real general matrix (Crout’s method) 388 LCX A system of linear equations with a complex general matrix (Crout’s method) 407 LSX LSIX LSBX LSBIX LBX1 LSTX LTX LAXR LCXR LSXR LSIXR LSBXR LBX1R Matrix inversion A system of linear equations with a positive-definite symmetric matrix (Modified Cholesky’s method) A system of linear equations with a real indefinite symmetric matrix (Block diagonal pivoting method) A system of linear equations with a positive-definite symmetric band matrix (Modified Cholesky’s method) A system of linear equations with a real indefinite symmetric band matrix (block diagonal pivoting method) A system of linear equations with a real general band matrix (Gaussian elimination method) A system of linear equations with a positive-definite symmetric tridiagonal matrix (Modified Cholesky’s method) A system of linear equations with a real tridiagonal matrix (Gaussian elimination method) Iterative refinement of the solution to a system of linear equations with a real general matrix Iterative refinament of the solution to a system of linear equations with a complex general matrix Iterative refinement of the solution to a system of linear equations with a positive-definite symmetric matrix Iterative refinament of the solution to a system of linear equations with a real indefinite symmetric matrix Iterative refinament of the solution to a system of linear equations with a positive-definite symmetric band matrix Iterative refinement of the solution to a system of linear equations with a real general band matrix Subroutine name Item Page Remarks LUIV The inverse of a real general matrix decomposed into the factors L and U 452 CLUIV The inverse of a complex general matrix decomposed into the factors L and U 279 LDIV Decomposition of matrices The inverse of a positive-definite symmetric matrix decomposed into the factors L, D and L T Subroutine name Item Page Remarks ALU LU-decomposition of a real general matrix (Crout’s method) 98 CLU LU-decomposition of a complex general matrix (Crout’s method) 277 SLDL SMDM SBDL SBMDM LDL T -decomposition of a positive-definite symmetric matrix (Modified Cholesky’s method) MDM T -decomposition of a real indefinite symmetric matrix (Block diagonal pivoting method) LDL T -decomposition of a positive-definite symmetric band matrix (Modified Cholesky’s method) MDM T -decomposition of a real indefinite symmetric band matrix (block diagonal pivoting method) BLU1 LU-decomposition of a real general band matrix (Gaussian elimination method) 189 445 438 433 431 402 442 449 399 409 447 440 435 404 412 570 572 553 555
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 13 and 14: Eigenvalues 4 Subroutine name Item
Page 15 and 16: Approximation 6 Subroutine name Ite
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
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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,
Page 462 and 463:
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
Page 498 and 499:
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
Page 508 and 509:
500 FORMAT(I5) 510 FORMAT(5E15.7) 6
Page 510 and 511:
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)
Page 518 and 519:
H11-20-0151 ODGE, DODGE A stiff sys
Page 520 and 521:
Table ODGE-1 Condition codes Code M
Page 522 and 523:
Method Both Gear’s and Adams meth
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xm+1 is determined, solution ye at
Page 526 and 527:
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
Page 538 and 539:
2) For u − 0 . 5 > 0. 46875 G -1
Page 540 and 541:
J12-10-0101 RANP2 Generation of Poi
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J11-10-0101 RANU2 Generation of uni
Page 544 and 545:
There is a theory that such a perio
Page 546 and 547:
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
Page 552 and 553:
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
Page 562 and 563:
A52-31-0202 SBDL, DSBDL LDL T -deco
Page 564 and 565:
A52-21-0202 SBMDM, DSBMDM MDM T dec
Page 566 and 567:
1) Determines 2) If ( k − 1) ( k
Page 568 and 569:
DO 10 I=1,N WRITE(6,630) I,(EV(I,J)
Page 570 and 571:
eigenvectors are determined using t
Page 572 and 573:
A21-11-0201 SGGM, DSGGM Subtraction
Page 574 and 575:
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
Page 586 and 587:
E31-21-0101 SMLE2, DSMLE2 Data smoo
Page 588 and 589:
E11-21-0101 SPLV, DSPLV Cubic splin
Page 590 and 591:
(c) (ISW(1) =3,DY (1)=, f"( x1)/f"(
Page 592 and 593:
B21-21-0602 TEIG1, DTEIG1 Eigenvalu
Page 594 and 595:
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
Page 602 and 603:
B21-25-0302 TRIDH, DTRIDH Reduction
Page 604 and 605:
where, l = n − k + 1 1 2 1 = k tl
Page 606 and 607:
( ( ) ) ( ( ) ) ( ( ) ) 2 σ k k 2
Page 608 and 609:
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
Page 616 and 617:
APPENDIX A AUXILIARY SUBROUTINES A.
Page 618 and 619:
A.3 MGSSL Printing of condition mes
Page 620 and 621:
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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