SUBROUTINE DQRDC(X,LDX,N,P,QRAUX,JPVT,WORK,JOB) INTEGER LDX,N,P,JOB INTEGER JPVT(1) DOUBLE PRECISION X(LDX,1),QRAUX(1),WORK(1) C C DQRDC USES HOUSEHOLDER TRANSFORMATIONS TO COMPUTE THE QR C FACTORIZATION OF AN N BY P MATRIX X. COLUMN PIVOTING C BASED ON THE 2-NORMS OF THE REDUCED COLUMNS MAY BE C PERFORMED AT THE USERS OPTION. C C ON ENTRY C C X DOUBLE PRECISION(LDX,P), WHERE LDX .GE. N. C X CONTAINS THE MATRIX WHOSE DECOMPOSITION IS TO BE C COMPUTED. C C LDX INTEGER. C LDX IS THE LEADING DIMENSION OF THE ARRAY X. C C N INTEGER. C N IS THE NUMBER OF ROWS OF THE MATRIX X. C C P INTEGER. C P IS THE NUMBER OF COLUMNS OF THE MATRIX X. C C JPVT INTEGER(P). C JPVT CONTAINS INTEGERS THAT CONTROL THE SELECTION C OF THE PIVOT COLUMNS. THE K-TH COLUMN X(K) OF X C IS PLACED IN ONE OF THREE CLASSES ACCORDING TO THE C VALUE OF JPVT(K). C C IF JPVT(K) .GT. 0, THEN X(K) IS AN INITIAL C COLUMN. C C IF JPVT(K) .EQ. 0, THEN X(K) IS A FREE COLUMN. C C IF JPVT(K) .LT. 0, THEN X(K) IS A FINAL COLUMN. C C BEFORE THE DECOMPOSITION IS COMPUTED, INITIAL COLUMNS C ARE MOVED TO THE BEGINNING OF THE ARRAY X AND FINAL C COLUMNS TO THE END. BOTH INITIAL AND FINAL COLUMNS C ARE FROZEN IN PLACE DURING THE COMPUTATION AND ONLY C FREE COLUMNS ARE MOVED. AT THE K-TH STAGE OF THE C REDUCTION, IF X(K) IS OCCUPIED BY A FREE COLUMN C IT IS INTERCHANGED WITH THE FREE COLUMN OF LARGEST C REDUCED NORM. JPVT IS NOT REFERENCED IF C JOB .EQ. 0. C C WORK DOUBLE PRECISION(P). C WORK IS A WORK ARRAY. WORK IS NOT REFERENCED IF C JOB .EQ. 0. C C JOB INTEGER. C JOB IS AN INTEGER THAT INITIATES COLUMN PIVOTING. C IF JOB .EQ. 0, NO PIVOTING IS DONE. C IF JOB .NE. 0, PIVOTING IS DONE. C C ON RETURN C C X X CONTAINS IN ITS UPPER TRIANGLE THE UPPER C TRIANGULAR MATRIX R OF THE QR FACTORIZATION. C BELOW ITS DIAGONAL X CONTAINS INFORMATION FROM C WHICH THE ORTHOGONAL PART OF THE DECOMPOSITION C CAN BE RECOVERED. NOTE THAT IF PIVOTING HAS C BEEN REQUESTED, THE DECOMPOSITION IS NOT THAT C OF THE ORIGINAL MATRIX X BUT THAT OF X C WITH ITS COLUMNS PERMUTED AS DESCRIBED BY JPVT. C C QRAUX DOUBLE PRECISION(P). C QRAUX CONTAINS FURTHER INFORMATION REQUIRED TO RECOVER C THE ORTHOGONAL PART OF THE DECOMPOSITION. C C JPVT JPVT(K) CONTAINS THE INDEX OF THE COLUMN OF THE C ORIGINAL MATRIX THAT HAS BEEN INTERCHANGED INTO C THE K-TH COLUMN, IF PIVOTING WAS REQUESTED. C C LINPACK. THIS VERSION DATED 08/14/78 . C G.W. STEWART, UNIVERSITY OF MARYLAND, ARGONNE NATIONAL LAB. C C DQRDC USES THE FOLLOWING FUNCTIONS AND SUBPROGRAMS. C C BLAS DAXPY,DDOT,DSCAL,DSWAP,DNRM2 C FORTRAN DABS,DMAX1,MIN0,DSQRT C C INTERNAL VARIABLES C INTEGER J,JP,L,LP1,LUP,MAXJ,PL,PU DOUBLE PRECISION MAXNRM,DNRM2,TT DOUBLE PRECISION DDOT,NRMXL,T LOGICAL NEGJ,SWAPJ C C PL = 1 PU = 0 IF (JOB .EQ. 0) GO TO 60 C C PIVOTING HAS BEEN REQUESTED. REARRANGE THE COLUMNS C ACCORDING TO JPVT. C DO 20 J = 1, P SWAPJ = JPVT(J) .GT. 0 NEGJ = JPVT(J) .LT. 0 JPVT(J) = J IF (NEGJ) JPVT(J) = -J IF (.NOT.SWAPJ) GO TO 10 IF (J .NE. PL) CALL DSWAP(N,X(1,PL),1,X(1,J),1) JPVT(J) = JPVT(PL) JPVT(PL) = J PL = PL + 1 10 CONTINUE 20 CONTINUE PU = P DO 50 JJ = 1, P J = P - JJ + 1 IF (JPVT(J) .GE. 0) GO TO 40 JPVT(J) = -JPVT(J) IF (J .EQ. PU) GO TO 30 CALL DSWAP(N,X(1,PU),1,X(1,J),1) JP = JPVT(PU) JPVT(PU) = JPVT(J) JPVT(J) = JP 30 CONTINUE PU = PU - 1 40 CONTINUE 50 CONTINUE 60 CONTINUE C C COMPUTE THE NORMS OF THE FREE COLUMNS. C IF (PU .LT. PL) GO TO 80 DO 70 J = PL, PU QRAUX(J) = DNRM2(N,X(1,J),1) WORK(J) = QRAUX(J) 70 CONTINUE 80 CONTINUE C C PERFORM THE HOUSEHOLDER REDUCTION OF X. C LUP = MIN0(N,P) DO 200 L = 1, LUP IF (L .LT. PL .OR. L .GE. PU) GO TO 120 C C LOCATE THE COLUMN OF LARGEST NORM AND BRING IT C INTO THE PIVOT POSITION. C MAXNRM = 0.0D0 MAXJ = L DO 100 J = L, PU IF (QRAUX(J) .LE. MAXNRM) GO TO 90 MAXNRM = QRAUX(J) MAXJ = J 90 CONTINUE 100 CONTINUE IF (MAXJ .EQ. L) GO TO 110 CALL DSWAP(N,X(1,L),1,X(1,MAXJ),1) QRAUX(MAXJ) = QRAUX(L) WORK(MAXJ) = WORK(L) JP = JPVT(MAXJ) JPVT(MAXJ) = JPVT(L) JPVT(L) = JP 110 CONTINUE 120 CONTINUE QRAUX(L) = 0.0D0 IF (L .EQ. N) GO TO 190 C C COMPUTE THE HOUSEHOLDER TRANSFORMATION FOR COLUMN L. C NRMXL = DNRM2(N-L+1,X(L,L),1) IF (NRMXL .EQ. 0.0D0) GO TO 180 IF (X(L,L) .NE. 0.0D0) NRMXL = DSIGN(NRMXL,X(L,L)) CALL DSCAL(N-L+1,1.0D0/NRMXL,X(L,L),1) X(L,L) = 1.0D0 + X(L,L) C C APPLY THE TRANSFORMATION TO THE REMAINING COLUMNS, C UPDATING THE NORMS. C LP1 = L + 1 IF (P .LT. LP1) GO TO 170 DO 160 J = LP1, P T = -DDOT(N-L+1,X(L,L),1,X(L,J),1)/X(L,L) CALL DAXPY(N-L+1,T,X(L,L),1,X(L,J),1) IF (J .LT. PL .OR. J .GT. PU) GO TO 150 IF (QRAUX(J) .EQ. 0.0D0) GO TO 150 TT = 1.0D0 - (DABS(X(L,J))/QRAUX(J))**2 TT = DMAX1(TT,0.0D0) T = TT TT = 1.0D0 + 0.05D0*TT*(QRAUX(J)/WORK(J))**2 IF (TT .EQ. 1.0D0) GO TO 130 QRAUX(J) = QRAUX(J)*DSQRT(T) GO TO 140 130 CONTINUE QRAUX(J) = DNRM2(N-L,X(L+1,J),1) WORK(J) = QRAUX(J) 140 CONTINUE 150 CONTINUE 160 CONTINUE 170 CONTINUE C C SAVE THE TRANSFORMATION. C QRAUX(L) = X(L,L) X(L,L) = -NRMXL 180 CONTINUE 190 CONTINUE 200 CONTINUE RETURN END SUBROUTINE DSVDC(X,LDX,N,P,S,E,U,LDU,V,LDV,WORK,JOB,INFO) INTEGER LDX,N,P,LDU,LDV,JOB,INFO DOUBLE PRECISION X(LDX,1),S(1),E(1),U(LDU,1),V(LDV,1),WORK(1) C C C DSVDC IS A SUBROUTINE TO REDUCE A DOUBLE PRECISION NXP MATRIX X C BY ORTHOGONAL TRANSFORMATIONS U AND V TO DIAGONAL FORM. THE C DIAGONAL ELEMENTS S(I) ARE THE SINGULAR VALUES OF X. THE C COLUMNS OF U ARE THE CORRESPONDING LEFT SINGULAR VECTORS, C AND THE COLUMNS OF V THE RIGHT SINGULAR VECTORS. C C ON ENTRY C C X DOUBLE PRECISION(LDX,P), WHERE LDX.GE.N. C X CONTAINS THE MATRIX WHOSE SINGULAR VALUE C DECOMPOSITION IS TO BE COMPUTED. X IS C DESTROYED BY DSVDC. C C LDX INTEGER. C LDX IS THE LEADING DIMENSION OF THE ARRAY X. C C N INTEGER. C N IS THE NUMBER OF COLUMNS OF THE MATRIX X. C C P INTEGER. C P IS THE NUMBER OF ROWS OF THE MATRIX X. C C LDU INTEGER. C LDU IS THE LEADING DIMENSION OF THE ARRAY U. C (SEE BELOW). C C LDV INTEGER. C LDV IS THE LEADING DIMENSION OF THE ARRAY V. C (SEE BELOW). C C WORK DOUBLE PRECISION(N). C WORK IS A SCRATCH ARRAY. C C JOB INTEGER. C JOB CONTROLS THE COMPUTATION OF THE SINGULAR C VECTORS. IT HAS THE DECIMAL EXPANSION AB C WITH THE FOLLOWING MEANING C C A.EQ.0 DO NOT COMPUTE THE LEFT SINGULAR C VECTORS. C A.EQ.1 RETURN THE N LEFT SINGULAR VECTORS C IN U. C A.GE.2 RETURN THE FIRST MIN(N,P) SINGULAR C VECTORS IN U. C B.EQ.0 DO NOT COMPUTE THE RIGHT SINGULAR C VECTORS. C B.EQ.1 RETURN THE RIGHT SINGULAR VECTORS C IN V. C C ON RETURN C C S DOUBLE PRECISION(MM), WHERE MM=MIN(N+1,P). C THE FIRST MIN(N,P) ENTRIES OF S CONTAIN THE C SINGULAR VALUES OF X ARRANGED IN DESCENDING C ORDER OF MAGNITUDE. C C E DOUBLE PRECISION(P). C E ORDINARILY CONTAINS ZEROS. HOWEVER SEE THE C DISCUSSION OF INFO FOR EXCEPTIONS. C C U DOUBLE PRECISION(LDU,K), WHERE LDU.GE.N. IF C JOBA.EQ.1 THEN K.EQ.N, IF JOBA.GE.2 C THEN K.EQ.MIN(N,P). C U CONTAINS THE MATRIX OF RIGHT SINGULAR VECTORS. C U IS NOT REFERENCED IF JOBA.EQ.0. IF N.LE.P C OR IF JOBA.EQ.2, THEN U MAY BE IDENTIFIED WITH X C IN THE SUBROUTINE CALL. C C V DOUBLE PRECISION(LDV,P), WHERE LDV.GE.P. C V CONTAINS THE MATRIX OF RIGHT SINGULAR VECTORS. C V IS NOT REFERENCED IF JOB.EQ.0. IF P.LE.N, C THEN V MAY BE IDENTIFIED WITH X IN THE C SUBROUTINE CALL. C C INFO INTEGER. C THE SINGULAR VALUES (AND THEIR CORRESPONDING C SINGULAR VECTORS) S(INFO+1),S(INFO+2),...,S(M) C ARE CORRECT (HERE M=MIN(N,P)). THUS IF C INFO.EQ.0, ALL THE SINGULAR VALUES AND THEIR C VECTORS ARE CORRECT. IN ANY EVENT, THE MATRIX C B = TRANS(U)*X*V IS THE BIDIAGONAL MATRIX C WITH THE ELEMENTS OF S ON ITS DIAGONAL AND THE C ELEMENTS OF E ON ITS SUPER-DIAGONAL (TRANS(U) C IS THE TRANSPOSE OF U). THUS THE SINGULAR C VALUES OF X AND B ARE THE SAME. C C LINPACK. THIS VERSION DATED 03/19/79 . C G.W. STEWART, UNIVERSITY OF MARYLAND, ARGONNE NATIONAL LAB. C C DSVDC USES THE FOLLOWING FUNCTIONS AND SUBPROGRAMS. C C EXTERNAL DROT C BLAS DAXPY,DDOT,DSCAL,DSWAP,DNRM2,DROTG C FORTRAN DABS,DMAX1,MAX0,MIN0,MOD,DSQRT C C INTERNAL VARIABLES C INTEGER I,ITER,J,JOBU,K,KASE,KK,L,LL,LLS,LM1,LP1,LS,LU,M,MAXIT, * MM,MM1,MP1,NCT,NCTP1,NCU,NRT,NRTP1 DOUBLE PRECISION DDOT,T DOUBLE PRECISION B,C,CS,EL,EMM1,F,G,DNRM2,SCALE,SHIFT,SL,SM,SN, * SMM1,T1,TEST,ZTEST LOGICAL WANTU,WANTV C C C SET THE MAXIMUM NUMBER OF ITERATIONS. C MAXIT = 30 C C DETERMINE WHAT IS TO BE COMPUTED. C WANTU = .FALSE. WANTV = .FALSE. JOBU = MOD(JOB,100)/10 NCU = N IF (JOBU .GT. 1) NCU = MIN0(N,P) IF (JOBU .NE. 0) WANTU = .TRUE. IF (MOD(JOB,10) .NE. 0) WANTV = .TRUE. C C REDUCE X TO BIDIAGONAL FORM, STORING THE DIAGONAL ELEMENTS C IN S AND THE SUPER-DIAGONAL ELEMENTS IN E. C INFO = 0 NCT = MIN0(N-1,P) NRT = MAX0(0,MIN0(P-2,N)) LU = MAX0(NCT,NRT) IF (LU .LT. 1) GO TO 170 DO 160 L = 1, LU LP1 = L + 1 IF (L .GT. NCT) GO TO 20 C C COMPUTE THE TRANSFORMATION FOR THE L-TH COLUMN AND C PLACE THE L-TH DIAGONAL IN S(L). C S(L) = DNRM2(N-L+1,X(L,L),1) IF (S(L) .EQ. 0.0D0) GO TO 10 IF (X(L,L) .NE. 0.0D0) S(L) = DSIGN(S(L),X(L,L)) CALL DSCAL(N-L+1,1.0D0/S(L),X(L,L),1) X(L,L) = 1.0D0 + X(L,L) 10 CONTINUE S(L) = -S(L) 20 CONTINUE IF (P .LT. LP1) GO TO 50 DO 40 J = LP1, P IF (L .GT. NCT) GO TO 30 IF (S(L) .EQ. 0.0D0) GO TO 30 C C APPLY THE TRANSFORMATION. C T = -DDOT(N-L+1,X(L,L),1,X(L,J),1)/X(L,L) CALL DAXPY(N-L+1,T,X(L,L),1,X(L,J),1) 30 CONTINUE C C PLACE THE L-TH ROW OF X INTO E FOR THE C SUBSEQUENT CALCULATION OF THE ROW TRANSFORMATION. C E(J) = X(L,J) 40 CONTINUE 50 CONTINUE IF (.NOT.WANTU .OR. L .GT. NCT) GO TO 70 C C PLACE THE TRANSFORMATION IN U FOR SUBSEQUENT BACK C MULTIPLICATION. C DO 60 I = L, N U(I,L) = X(I,L) 60 CONTINUE 70 CONTINUE IF (L .GT. NRT) GO TO 150 C C COMPUTE THE L-TH ROW TRANSFORMATION AND PLACE THE C L-TH SUPER-DIAGONAL IN E(L). C E(L) = DNRM2(P-L,E(LP1),1) IF (E(L) .EQ. 0.0D0) GO TO 80 IF (E(LP1) .NE. 0.0D0) E(L) = DSIGN(E(L),E(LP1)) CALL DSCAL(P-L,1.0D0/E(L),E(LP1),1) E(LP1) = 1.0D0 + E(LP1) 80 CONTINUE E(L) = -E(L) IF (LP1 .GT. N .OR. E(L) .EQ. 0.0D0) GO TO 120 C C APPLY THE TRANSFORMATION. C DO 90 I = LP1, N WORK(I) = 0.0D0 90 CONTINUE DO 100 J = LP1, P CALL DAXPY(N-L,E(J),X(LP1,J),1,WORK(LP1),1) 100 CONTINUE DO 110 J = LP1, P CALL DAXPY(N-L,-E(J)/E(LP1),WORK(LP1),1,X(LP1,J),1) 110 CONTINUE 120 CONTINUE IF (.NOT.WANTV) GO TO 140 C C PLACE THE TRANSFORMATION IN V FOR SUBSEQUENT C BACK MULTIPLICATION. C DO 130 I = LP1, P V(I,L) = E(I) 130 CONTINUE 140 CONTINUE 150 CONTINUE 160 CONTINUE 170 CONTINUE C C SET UP THE FINAL BIDIAGONAL MATRIX OR ORDER M. C M = MIN0(P,N+1) NCTP1 = NCT + 1 NRTP1 = NRT + 1 IF (NCT .LT. P) S(NCTP1) = X(NCTP1,NCTP1) IF (N .LT. M) S(M) = 0.0D0 IF (NRTP1 .LT. M) E(NRTP1) = X(NRTP1,M) E(M) = 0.0D0 C C IF REQUIRED, GENERATE U. C IF (.NOT.WANTU) GO TO 300 IF (NCU .LT. NCTP1) GO TO 200 DO 190 J = NCTP1, NCU DO 180 I = 1, N U(I,J) = 0.0D0 180 CONTINUE U(J,J) = 1.0D0 190 CONTINUE 200 CONTINUE IF (NCT .LT. 1) GO TO 290 DO 280 LL = 1, NCT L = NCT - LL + 1 IF (S(L) .EQ. 0.0D0) GO TO 250 LP1 = L + 1 IF (NCU .LT. LP1) GO TO 220 DO 210 J = LP1, NCU T = -DDOT(N-L+1,U(L,L),1,U(L,J),1)/U(L,L) CALL DAXPY(N-L+1,T,U(L,L),1,U(L,J),1) 210 CONTINUE 220 CONTINUE CALL DSCAL(N-L+1,-1.0D0,U(L,L),1) U(L,L) = 1.0D0 + U(L,L) LM1 = L - 1 IF (LM1 .LT. 1) GO TO 240 DO 230 I = 1, LM1 U(I,L) = 0.0D0 230 CONTINUE 240 CONTINUE GO TO 270 250 CONTINUE DO 260 I = 1, N U(I,L) = 0.0D0 260 CONTINUE U(L,L) = 1.0D0 270 CONTINUE 280 CONTINUE 290 CONTINUE 300 CONTINUE C C IF IT IS REQUIRED, GENERATE V. C IF (.NOT.WANTV) GO TO 350 DO 340 LL = 1, P L = P - LL + 1 LP1 = L + 1 IF (L .GT. NRT) GO TO 320 IF (E(L) .EQ. 0.0D0) GO TO 320 DO 310 J = LP1, P T = -DDOT(P-L,V(LP1,L),1,V(LP1,J),1)/V(LP1,L) CALL DAXPY(P-L,T,V(LP1,L),1,V(LP1,J),1) 310 CONTINUE 320 CONTINUE DO 330 I = 1, P V(I,L) = 0.0D0 330 CONTINUE V(L,L) = 1.0D0 340 CONTINUE 350 CONTINUE C C MAIN ITERATION LOOP FOR THE SINGULAR VALUES. C MM = M ITER = 0 360 CONTINUE C C QUIT IF ALL THE SINGULAR VALUES HAVE BEEN FOUND. C C ...EXIT IF (M .EQ. 0) GO TO 620 C C IF TOO MANY ITERATIONS HAVE BEEN PERFORMED, SET C FLAG AND RETURN. C IF (ITER .LT. MAXIT) GO TO 370 INFO = M C ......EXIT GO TO 620 370 CONTINUE C C THIS SECTION OF THE PROGRAM INSPECTS FOR C NEGLIGIBLE ELEMENTS IN THE S AND E ARRAYS. ON C COMPLETION THE VARIABLES KASE AND L ARE SET AS FOLLOWS. C C KASE = 1 IF S(M) AND E(L-1) ARE NEGLIGIBLE AND L.LT.M C KASE = 2 IF S(L) IS NEGLIGIBLE AND L.LT.M C KASE = 3 IF E(L-1) IS NEGLIGIBLE, L.LT.M, AND C S(L), ..., S(M) ARE NOT NEGLIGIBLE (QR STEP). C KASE = 4 IF E(M-1) IS NEGLIGIBLE (CONVERGENCE). C DO 390 LL = 1, M L = M - LL C ...EXIT IF (L .EQ. 0) GO TO 400 TEST = DABS(S(L)) + DABS(S(L+1)) ZTEST = TEST + DABS(E(L)) IF (ZTEST .NE. TEST) GO TO 380 E(L) = 0.0D0 C ......EXIT GO TO 400 380 CONTINUE 390 CONTINUE 400 CONTINUE IF (L .NE. M - 1) GO TO 410 KASE = 4 GO TO 480 410 CONTINUE LP1 = L + 1 MP1 = M + 1 DO 430 LLS = LP1, MP1 LS = M - LLS + LP1 C ...EXIT IF (LS .EQ. L) GO TO 440 TEST = 0.0D0 IF (LS .NE. M) TEST = TEST + DABS(E(LS)) IF (LS .NE. L + 1) TEST = TEST + DABS(E(LS-1)) ZTEST = TEST + DABS(S(LS)) IF (ZTEST .NE. TEST) GO TO 420 S(LS) = 0.0D0 C ......EXIT GO TO 440 420 CONTINUE 430 CONTINUE 440 CONTINUE IF (LS .NE. L) GO TO 450 KASE = 3 GO TO 470 450 CONTINUE IF (LS .NE. M) GO TO 460 KASE = 1 GO TO 470 460 CONTINUE KASE = 2 L = LS 470 CONTINUE 480 CONTINUE L = L + 1 C C PERFORM THE TASK INDICATED BY KASE. C GO TO (490,520,540,570), KASE C C DEFLATE NEGLIGIBLE S(M). C 490 CONTINUE MM1 = M - 1 F = E(M-1) E(M-1) = 0.0D0 DO 510 KK = L, MM1 K = MM1 - KK + L T1 = S(K) CALL DROTG(T1,F,CS,SN) S(K) = T1 IF (K .EQ. L) GO TO 500 F = -SN*E(K-1) E(K-1) = CS*E(K-1) 500 CONTINUE IF (WANTV) CALL DROT(P,V(1,K),1,V(1,M),1,CS,SN) 510 CONTINUE GO TO 610 C C SPLIT AT NEGLIGIBLE S(L). C 520 CONTINUE F = E(L-1) E(L-1) = 0.0D0 DO 530 K = L, M T1 = S(K) CALL DROTG(T1,F,CS,SN) S(K) = T1 F = -SN*E(K) E(K) = CS*E(K) IF (WANTU) CALL DROT(N,U(1,K),1,U(1,L-1),1,CS,SN) 530 CONTINUE GO TO 610 C C PERFORM ONE QR STEP. C 540 CONTINUE C C CALCULATE THE SHIFT. C SCALE = DMAX1(DABS(S(M)),DABS(S(M-1)),DABS(E(M-1)), * DABS(S(L)),DABS(E(L))) SM = S(M)/SCALE SMM1 = S(M-1)/SCALE EMM1 = E(M-1)/SCALE SL = S(L)/SCALE EL = E(L)/SCALE B = ((SMM1 + SM)*(SMM1 - SM) + EMM1**2)/2.0D0 C = (SM*EMM1)**2 SHIFT = 0.0D0 IF (B .EQ. 0.0D0 .AND. C .EQ. 0.0D0) GO TO 550 SHIFT = DSQRT(B**2+C) IF (B .LT. 0.0D0) SHIFT = -SHIFT SHIFT = C/(B + SHIFT) 550 CONTINUE C C I changed this line from - shift to + shift C This bug was communicated by Ake Bjork C Mike Meyer jan 9 1984 C F = (SL + SM)*(SL - SM) - SHIFT C F = (SL + SM)*(SL - SM) + SHIFT G = SL*EL C C CHASE ZEROS. C MM1 = M - 1 DO 560 K = L, MM1 CALL DROTG(F,G,CS,SN) IF (K .NE. L) E(K-1) = F F = CS*S(K) + SN*E(K) E(K) = CS*E(K) - SN*S(K) G = SN*S(K+1) S(K+1) = CS*S(K+1) IF (WANTV) CALL DROT(P,V(1,K),1,V(1,K+1),1,CS,SN) CALL DROTG(F,G,CS,SN) S(K) = F F = CS*E(K) + SN*S(K+1) S(K+1) = -SN*E(K) + CS*S(K+1) G = SN*E(K+1) E(K+1) = CS*E(K+1) IF (WANTU .AND. K .LT. N) * CALL DROT(N,U(1,K),1,U(1,K+1),1,CS,SN) 560 CONTINUE E(M-1) = F ITER = ITER + 1 GO TO 610 C C CONVERGENCE. C 570 CONTINUE C C MAKE THE SINGULAR VALUE POSITIVE. C IF (S(L) .GE. 0.0D0) GO TO 580 S(L) = -S(L) IF (WANTV) CALL DSCAL(P,-1.0D0,V(1,L),1) 580 CONTINUE C C ORDER THE SINGULAR VALUE. C 590 IF (L .EQ. MM) GO TO 600 C ...EXIT IF (S(L) .GE. S(L+1)) GO TO 600 T = S(L) S(L) = S(L+1) S(L+1) = T IF (WANTV .AND. L .LT. P) * CALL DSWAP(P,V(1,L),1,V(1,L+1),1) IF (WANTU .AND. L .LT. N) * CALL DSWAP(N,U(1,L),1,U(1,L+1),1) L = L + 1 GO TO 590 600 CONTINUE ITER = 0 M = M - 1 610 CONTINUE GO TO 360 620 CONTINUE RETURN END