SSL II USER'S GUIDE - Lahey Computer Systems

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

Table 3.3 Subroutines for matrix manipulation B or x Real general Real symmetric Vector A matrix matrix Real general matrix Addition AGGM (A21-11-0101) Subtraction SGGM (A21-11-0301) Multiplication MGGM MGSM MAV (A21-11-0301) (A21-11-0401) (A21-13-0101) Complex general matrix Multiplication MCV (A21-15-0101) Real symmetric matrix Addition ASSM (A21-12-0101) Subtraction SSSM (A21-12-0201) Multiplication MSGM MSSM MSV (A21-12-0401) (A21-12-0301) (A21-14-0101) Real general band matrix Multiplication MBV (A51-11-0101) Real symmetric band matrix Multiplication MSBV (A51-14-0101) • Addition/Subtraction of two matrices A ± B • Multiplication of a matrix by a vector Ax • Multiplication of two matrices AB SSL II provides the subroutines for matrix manipulation, as listed in Table 3.3. Comments on use These subroutines for multiplication of matrix by vector are designed to obtain the residual vector as well, so that the subroutines can be used for iterative methods for linear equations. 3.4 LINEAR EQUATIONS AND MATRIX INVERSION (DIRECT METHOD) This section describes the subroutines that is used to solve the following problems. • Solve systems of linear equations Ax = b A is an n × n matrix, x and b are n-dimensional vectors. • Obtain the inverse of a matrix A. • Obtain the determinant of a matrix A. In order to solve the above problems, SSL II provides the following basic subroutines ( here we call them component subroutines) for each matrix structure. LINEAR ALGEBRA (a) Numeric decomposition of a coefficient matrix (b) Solving based on the decomposed coefficient matrix (c) Iterative refinement of the initial solution (d) Matrix inversion based on the decomposed matrix Combinations of the subroutines ensure that systems of linear equations, inverse matrices, and the determinants can be obtained. • Linear equations The solution of the equations can be obtained by calling the component routines consecutively as follows: : CALL Component routine from (a) CALL Component routine from (b) : • Matrix inversion The inverse can be obtained by calling the above components routines serially as follows: : CALL Component routine from (a) CALL Component routine from (b) : The inverse of band matrices generally result in dense matrices so that to obtain such the inverse is not beneficial. That is why those component routines for the inverse are not prepared in SSL II. 29
GENERAL DESCRIPTION • Determinants There is no component routine which computes the values of determinant. However, the values can be obtained from the elements resulting from decomposion (a). Though any problem can be solved by properly combining these component routines, SSL II also has routines, called standard routines, in which component routines are called internally. It is recommended that the user calls these standard routines. Table 3.4 lists the standard routines and component routines. Comments on uses • Iterative refinement of a solution In order to refine the accuracy of solution obtained by standard routines, component routines (c) should be successively called regarding the solution as the initial approximation. In addition, the component routine (c) serves to estimate the accuracy of the initial approximation. • Matrix inversion Usually, it is not advisable to invert a matrix when solving a system of linear equations. 30 Ax = b (3.1) That is, in solving equations (3.1), the solution should not be obtained by calculating the inverse A -1 and then multiplying b by A -1 from the left side as shown in (3.2). x = A -1 b (3.2) Instead, it is advisable to compute the LUdecomposition of A and then perform the operations (forward and backward substitutions) shown in (3.3). Ly = b (3.3) Ux = y Higher operating speed and accuracy can be attained by using method (3.3). The approximate number of multiplications involved in the two methods (3.2) and (3.3) are compared in (3.4). For (3.2) For (3.3) 3 n + n 3 n / 3 2 Therefore, matrix inversion should only be performed when absolutely necessary. (3.4) • Equations with identical coefficient matrices When solving a number of systems of linear equations as in (3.5) where the coefficient matrices are the identical and the constant vectors are the different, Ax = b Ax Ax 1 2 m = b : 1 2 = b m ⎫ ⎪ ⎬ ⎪ ⎪ ⎭ (3.5) it is not advisable to decompose the coefficient A for each equation. After decomposing A when solving the first equation, only the forward and backward substitution shown in (3.3) should be performed for solving the other equations. In standard routines, a parameter ISW is provided for the user to control whether or not the equations is the first one to solve with the coefficient matrix. Notes and internal processing When using subroutines, the following should be noted from the viewpoint of internal processing. • Crout’s method, modified Cholesky’s method In SSL II, the Crout’s method is used for decomposing a general matrix. Generally the Gaussian elimination and the Crout’s method are known for decomposing (as in (3.6)) general matrices. A = LU (3.6) where: L is a lower triangular matrix and U is an upper triangular matrix. The two methods require the different calculation order, that is, the former is that intermediate values are calculated during decomposition and all elements are available only after decomposition. The latter is that each element is available during decomposition. Numerically the latter involves inner products for two vecters, so if that calculations can be performed precisely, the final decomposed matrices are more accurate than the former. Also, the data reference order in the Crout’s method is more localized during the decomposition process than in the Gaussian method. Therefore, in SSL II, the Crout’s method is used, and the inner products are carried out minimizing the effect of rounding errors. On the other hand, the modified Cholesky method is used for positive-definite symmetric matrices, that is, the decomposition shown in (3.7) is done.
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 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: CHAPTER 3 LINEAR ALGEBRA 3.1 OUTLIN
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
Page 69 and 70: GENERAL DESCRIPTION 60 y y ( λ,
Page 71 and 72: GENERAL DESCRIPTION 7.5 SERIES SSL
Page 73 and 74: GENERAL DESCRIPTION 64 which defini
Page 75 and 76: GENERAL DESCRIPTION 8.6 LAPLACE TRA
Page 77 and 78: GENERAL DESCRIPTION Since G(s) is r
Page 79 and 80: GENERAL DESCRIPTION CHAPTER 9 NUMER
Page 81 and 82: GENERAL DESCRIPTION Table 9.2 Numer
Page 83 and 84: GENERAL DESCRIPTION and those combi
Page 85 and 86: GENERAL DESCRIPTION • Final value
Page 87 and 88: 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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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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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)
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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
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from α α α x1 k x2 k x1 0 1 =
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faster than second order. • K pol
Page 558 and 559:
Letting ( ) λ 2 ( ) σ ( x) = x +
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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
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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
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A21-11-0201 SGGM, DSGGM Subtraction
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G23-11-0101 SIMP2, DSIMP2 Integrati
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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
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i.e., -s is stored in IP (k+1). The
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E31-11-0101 SMLE1, DSMLE1 Data smoo
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E31-21-0101 SMLE2, DSMLE2 Data smoo
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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
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• Direct sum of submatrices If T
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B21-21-0802 TRBK, DTRBK Back transf
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B21-25-0402 TRBKH, DTRBKH Back tran
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B21-25-0302 TRIDH, DTRIDH Reduction
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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−
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C23-11-0111 TSDM, DTSDM Zero of a r
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• Convergence criteria The follow
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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
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APPENDIX B ALPHABETIC GUIDE FOR SUB
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TableB.1-conteimued Subroutine Clas
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B.3 AUXILIARY SUBROUTINES Table B.3
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Interpolation and approximation Cod
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[20] Brezinski, C. Acceleration de
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[82] Uno,T. Bessel Function. Baifuk
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Author Subroutine Item T. Torii ECH
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