*DECK MINFIT SUBROUTINE MINFIT (NM, M, N, A, W, IP, B, IERR, RV1) C***BEGIN PROLOGUE MINFIT C***PURPOSE Compute the singular value decomposition of a rectangular C matrix and solve the related linear least squares problem. C***LIBRARY SLATEC (EISPACK) C***CATEGORY D9 C***TYPE SINGLE PRECISION (MINFIT-S) C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK C***AUTHOR Smith, B. T., et al. C***DESCRIPTION C C This subroutine is a translation of the ALGOL procedure MINFIT, C NUM. MATH. 14, 403-420(1970) by Golub and Reinsch. C HANDBOOK FOR AUTO. COMP., VOL II-LINEAR ALGEBRA, 134-151(1971). C C This subroutine determines, towards the solution of the linear C T C system AX=B, the singular value decomposition A=USV of a real C T C M by N rectangular matrix, forming U B rather than U. Householder C bidiagonalization and a variant of the QR algorithm are used. C C On INPUT C C NM must be set to the row dimension of the two-dimensional C array parameters, A and B, as declared in the calling C program dimension statement. Note that NM must be at least C as large as the maximum of M and N. NM is an INTEGER C variable. C C M is the number of rows of A and B. M is an INTEGER variable. C C N is the number of columns of A and the order of V. N is an C INTEGER variable. C C A contains the rectangular coefficient matrix of the system. C A is a two-dimensional REAL array, dimensioned A(NM,N). C C IP is the number of columns of B. IP can be zero. C C B contains the constant column matrix of the system if IP is C not zero. Otherwise, B is not referenced. B is a two- C dimensional REAL array, dimensioned B(NM,IP). C C On OUTPUT C C A has been overwritten by the matrix V (orthogonal) of the C decomposition in its first N rows and columns. If an C error exit is made, the columns of V corresponding to C indices of correct singular values should be correct. C C W contains the N (non-negative) singular values of A (the C diagonal elements of S). They are unordered. If an C error exit is made, the singular values should be correct C for indices IERR+1, IERR+2, ..., N. W is a one-dimensional C REAL array, dimensioned W(N). C C T C B has been overwritten by U B. If an error exit is made, C T C the rows of U B corresponding to indices of correct singular C values should be correct. C C IERR is an INTEGER flag set to C Zero for normal return, C K if the K-th singular value has not been C determined after 30 iterations. C The singular values should be correct for C indices IERR+1, IERR+2, ..., N. C C RV1 is a one-dimensional REAL array used for temporary storage, C dimensioned RV1(N). C C Calls PYTHAG(A,B) for sqrt(A**2 + B**2). C C Questions and comments should be directed to B. S. Garbow, C APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY C ------------------------------------------------------------------ C C***REFERENCES B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, C Y. Ikebe, V. C. Klema and C. B. Moler, Matrix Eigen- C system Routines - EISPACK Guide, Springer-Verlag, C 1976. C***ROUTINES CALLED PYTHAG C***REVISION HISTORY (YYMMDD) C 760101 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 890831 Modified array declarations. (WRB) C 890831 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C 920501 Reformatted the REFERENCES section. (WRB) C***END PROLOGUE MINFIT C INTEGER I,J,K,L,M,N,II,IP,I1,KK,K1,LL,L1,M1,NM,ITS,IERR REAL A(NM,*),W(*),B(NM,IP),RV1(*) REAL C,F,G,H,S,X,Y,Z,SCALE,S1 REAL PYTHAG C C***FIRST EXECUTABLE STATEMENT MINFIT IERR = 0 C .......... HOUSEHOLDER REDUCTION TO BIDIAGONAL FORM .......... G = 0.0E0 SCALE = 0.0E0 S1 = 0.0E0 C DO 300 I = 1, N L = I + 1 RV1(I) = SCALE * G G = 0.0E0 S = 0.0E0 SCALE = 0.0E0 IF (I .GT. M) GO TO 210 C DO 120 K = I, M 120 SCALE = SCALE + ABS(A(K,I)) C IF (SCALE .EQ. 0.0E0) GO TO 210 C DO 130 K = I, M A(K,I) = A(K,I) / SCALE S = S + A(K,I)**2 130 CONTINUE C F = A(I,I) G = -SIGN(SQRT(S),F) H = F * G - S A(I,I) = F - G IF (I .EQ. N) GO TO 160 C DO 150 J = L, N S = 0.0E0 C DO 140 K = I, M 140 S = S + A(K,I) * A(K,J) C F = S / H C DO 150 K = I, M A(K,J) = A(K,J) + F * A(K,I) 150 CONTINUE C 160 IF (IP .EQ. 0) GO TO 190 C DO 180 J = 1, IP S = 0.0E0 C DO 170 K = I, M 170 S = S + A(K,I) * B(K,J) C F = S / H C DO 180 K = I, M B(K,J) = B(K,J) + F * A(K,I) 180 CONTINUE C 190 DO 200 K = I, M 200 A(K,I) = SCALE * A(K,I) C 210 W(I) = SCALE * G G = 0.0E0 S = 0.0E0 SCALE = 0.0E0 IF (I .GT. M .OR. I .EQ. N) GO TO 290 C DO 220 K = L, N 220 SCALE = SCALE + ABS(A(I,K)) C IF (SCALE .EQ. 0.0E0) GO TO 290 C DO 230 K = L, N A(I,K) = A(I,K) / SCALE S = S + A(I,K)**2 230 CONTINUE C F = A(I,L) G = -SIGN(SQRT(S),F) H = F * G - S A(I,L) = F - G C DO 240 K = L, N 240 RV1(K) = A(I,K) / H C IF (I .EQ. M) GO TO 270 C DO 260 J = L, M S = 0.0E0 C DO 250 K = L, N 250 S = S + A(J,K) * A(I,K) C DO 260 K = L, N A(J,K) = A(J,K) + S * RV1(K) 260 CONTINUE C 270 DO 280 K = L, N 280 A(I,K) = SCALE * A(I,K) C 290 S1 = MAX(S1,ABS(W(I))+ABS(RV1(I))) 300 CONTINUE C .......... ACCUMULATION OF RIGHT-HAND TRANSFORMATIONS. C FOR I=N STEP -1 UNTIL 1 DO -- .......... DO 400 II = 1, N I = N + 1 - II IF (I .EQ. N) GO TO 390 IF (G .EQ. 0.0E0) GO TO 360 C DO 320 J = L, N C .......... DOUBLE DIVISION AVOIDS POSSIBLE UNDERFLOW .......... 320 A(J,I) = (A(I,J) / A(I,L)) / G C DO 350 J = L, N S = 0.0E0 C DO 340 K = L, N 340 S = S + A(I,K) * A(K,J) C DO 350 K = L, N A(K,J) = A(K,J) + S * A(K,I) 350 CONTINUE C 360 DO 380 J = L, N A(I,J) = 0.0E0 A(J,I) = 0.0E0 380 CONTINUE C 390 A(I,I) = 1.0E0 G = RV1(I) L = I 400 CONTINUE C IF (M .GE. N .OR. IP .EQ. 0) GO TO 510 M1 = M + 1 C DO 500 I = M1, N C DO 500 J = 1, IP B(I,J) = 0.0E0 500 CONTINUE C .......... DIAGONALIZATION OF THE BIDIAGONAL FORM .......... 510 CONTINUE C .......... FOR K=N STEP -1 UNTIL 1 DO -- .......... DO 700 KK = 1, N K1 = N - KK K = K1 + 1 ITS = 0 C .......... TEST FOR SPLITTING. C FOR L=K STEP -1 UNTIL 1 DO -- .......... 520 DO 530 LL = 1, K L1 = K - LL L = L1 + 1 IF (S1 + ABS(RV1(L)) .EQ. S1) GO TO 565 C .......... RV1(1) IS ALWAYS ZERO, SO THERE IS NO EXIT C THROUGH THE BOTTOM OF THE LOOP .......... IF (S1 + ABS(W(L1)) .EQ. S1) GO TO 540 530 CONTINUE C .......... CANCELLATION OF RV1(L) IF L GREATER THAN 1 .......... 540 C = 0.0E0 S = 1.0E0 C DO 560 I = L, K F = S * RV1(I) RV1(I) = C * RV1(I) IF (S1 + ABS(F) .EQ. S1) GO TO 565 G = W(I) H = PYTHAG(F,G) W(I) = H C = G / H S = -F / H IF (IP .EQ. 0) GO TO 560 C DO 550 J = 1, IP Y = B(L1,J) Z = B(I,J) B(L1,J) = Y * C + Z * S B(I,J) = -Y * S + Z * C 550 CONTINUE C 560 CONTINUE C .......... TEST FOR CONVERGENCE .......... 565 Z = W(K) IF (L .EQ. K) GO TO 650 C .......... SHIFT FROM BOTTOM 2 BY 2 MINOR .......... IF (ITS .EQ. 30) GO TO 1000 ITS = ITS + 1 X = W(L) Y = W(K1) G = RV1(K1) H = RV1(K) F = 0.5E0 * (((G + Z) / H) * ((G - Z) / Y) + Y / H - H / Y) G = PYTHAG(F,1.0E0) F = X - (Z / X) * Z + (H / X) * (Y / (F + SIGN(G,F)) - H) C .......... NEXT QR TRANSFORMATION .......... C = 1.0E0 S = 1.0E0 C DO 600 I1 = L, K1 I = I1 + 1 G = RV1(I) Y = W(I) H = S * G G = C * G Z = PYTHAG(F,H) RV1(I1) = Z C = F / Z S = H / Z F = X * C + G * S G = -X * S + G * C H = Y * S Y = Y * C C DO 570 J = 1, N X = A(J,I1) Z = A(J,I) A(J,I1) = X * C + Z * S A(J,I) = -X * S + Z * C 570 CONTINUE C Z = PYTHAG(F,H) W(I1) = Z C .......... ROTATION CAN BE ARBITRARY IF Z IS ZERO .......... IF (Z .EQ. 0.0E0) GO TO 580 C = F / Z S = H / Z 580 F = C * G + S * Y X = -S * G + C * Y IF (IP .EQ. 0) GO TO 600 C DO 590 J = 1, IP Y = B(I1,J) Z = B(I,J) B(I1,J) = Y * C + Z * S B(I,J) = -Y * S + Z * C 590 CONTINUE C 600 CONTINUE C RV1(L) = 0.0E0 RV1(K) = F W(K) = X GO TO 520 C .......... CONVERGENCE .......... 650 IF (Z .GE. 0.0E0) GO TO 700 C .......... W(K) IS MADE NON-NEGATIVE .......... W(K) = -Z C DO 690 J = 1, N 690 A(J,K) = -A(J,K) C 700 CONTINUE C GO TO 1001 C .......... SET ERROR -- NO CONVERGENCE TO A C SINGULAR VALUE AFTER 30 ITERATIONS .......... 1000 IERR = K 1001 RETURN END