SSL II USER'S GUIDE - Lahey Computer Systems

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

I. Special Functions Subroutine name Item Page Remarks CELI1 Complete elliptic Integral of the first kind K(x) 244 CELI2 Complete elliptic integral of the second kind E(x) 245 EXPI Exponentia l integral ( x) , E ( x) 304 Ei i SINI Sine integral Si (x) 569 COSI Cosine integral Ci (x) 285 SFRI Sine Fresnel integral S(x) 562 CFRI Cosine Fresnel integral C(x) 246 IGAM1 Incomplete Gamma function of the first kind γ (ν , x) 372 IGAM2 Incomplete Gamma function of the second kind Γ (ν , x) 373 IERF Inverse error function erf -1 (x) 369 IERFC Inverse complimented error function erfc -1 (x) 370 BJ0 Zero-order Bessel function of the first kind J0(x) 173 BJ1 First-order Bessel function of the first kind J1(x) 175 BY0 Zero-order Bessel function of the second kind Y0(x) 228 BY1 First-order Bessel function of the second kind Y1(x) 230 BI0 Modified Zero-order Bessel function of the first kind I0(x) 167 BI1 Modified First-order Bessel function of the first kind I1(x) 168 BK0 Modified Zero-order Bessel function of the second kind K0(x) 182 BK1 Modified First-order Bessel function of the second kind K1(x) 183 BJN Nth-order Bessel function of the first kind Jn (x) 169 BYN Nth-order Bessel function of the second kind Yn (x) 223 BIN Modified Nth-order Bessel function of the first kind In (x) 164 BKN Modified Nth-order Bessel function of the second kind Kn (x) 177 CBIN Modified Nth-order Bessel function of the first kind In (z) with complex variable 232 CBKN Modified Nth-order Bessel function of the second kind Kn (z) with complex variable 236 CBJN Integer order Bessel function of the first kind with complex variable Jn (z) 233 CBYN Integer order Bessel function of the second kind with complex variable Yn (z) 241 BJR Real-order Bessel function of the first kind Jν (x) 171 BYR Real-order Bessel function of the second kind Yν (x) 224 BIR Modified real-order Bessel function of the first kind Iν (x) 166 BKR Real order modified Bessel function of the second kind Kν (x) 178 CBJR Real-order Bessel function of the first kind with a complex variable Jν (z) 234 NDF Normal distribution function φ (x) 480 NDFC Complementary normal distribution function ψ (x) 481 INDF Inverse normal distribution function φ -1 (x) 375 INDFC Inverse complementary normal distribution function ψ -1 (x) 376 9
J. Pseudo Random Numbers 10 Subroutine name Item Page Remarks RANU2 Uniform (0, 1) pseudo random numbers 533 # RANU3 Shuffled uniform (0, 1) pseudo random numbers 536 # RANN1 Fast normal pseudo random numbers 528 # RANN2 Normal pseudo random numbers 530 # RANE2 Exponential pseudo random numbers 527 # RANP2 Poisson pseudo random integers 531 # RANB2 Binominal pseudo random numbers 525 # RATF1 Frequency test for uniform (0, 1) pseudo random numbers 538 # RATR1 Run test of up-and-down for uniform (0, 1) pseudo random numbers 540 #
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 and 14: Eigenvalues 4 Subroutine name Item
Page 15 and 16: Approximation 6 Subroutine name Ite
Page 17: Numerical Quadrature 8 Subroutine n
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
Page 67 and 68: GENERAL DESCRIPTION 58 − Calculat
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GENERAL DESCRIPTION 60 y y ( λ,
Page 71 and 72:
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
Page 114 and 115:
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:
Page 158 and 159:
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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
Page 422 and 423:
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
Page 464 and 465:
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
Page 484 and 485:
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
Page 534 and 535:
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
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
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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
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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