*DECK DQRDC SUBROUTINE DQRDC (X, LDX, N, P, QRAUX, JPVT, WORK, JOB) C***BEGIN PROLOGUE DQRDC C***PURPOSE Use Householder transformations to compute the QR C factorization of an N by P matrix. Column pivoting is a C users option. C***LIBRARY SLATEC (LINPACK) C***CATEGORY D5 C***TYPE DOUBLE PRECISION (SQRDC-S, DQRDC-D, CQRDC-C) C***KEYWORDS LINEAR ALGEBRA, LINPACK, MATRIX, ORTHOGONAL TRIANGULAR, C QR DECOMPOSITION C***AUTHOR Stewart, G. W., (U. of Maryland) C***DESCRIPTION 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 user's 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***REFERENCES J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W. C Stewart, LINPACK Users' Guide, SIAM, 1979. C***ROUTINES CALLED DAXPY, DDOT, DNRM2, DSCAL, DSWAP C***REVISION HISTORY (YYMMDD) C 780814 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 900326 Removed duplicate information from DESCRIPTION section. C (WRB) C 920501 Reformatted the REFERENCES section. (WRB) C***END PROLOGUE DQRDC INTEGER LDX,N,P,JOB INTEGER JPVT(*) DOUBLE PRECISION X(LDX,*),QRAUX(*),WORK(*) 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***FIRST EXECUTABLE STATEMENT DQRDC 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 = MIN(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 = SIGN(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 - (ABS(X(L,J))/QRAUX(J))**2 TT = MAX(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)*SQRT(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