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229 lines
8 KiB
Fortran
229 lines
8 KiB
Fortran
*DECK CQRDC
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SUBROUTINE CQRDC (X, LDX, N, P, QRAUX, JPVT, WORK, JOB)
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C***BEGIN PROLOGUE CQRDC
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C***PURPOSE Use Householder transformations to compute the QR
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C factorization of an N by P matrix. Column pivoting is a
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C users option.
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C***LIBRARY SLATEC (LINPACK)
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C***CATEGORY D5
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C***TYPE COMPLEX (SQRDC-S, DQRDC-D, CQRDC-C)
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C***KEYWORDS LINEAR ALGEBRA, LINPACK, MATRIX, ORTHOGONAL TRIANGULAR,
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C QR DECOMPOSITION
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C***AUTHOR Stewart, G. W., (U. of Maryland)
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C***DESCRIPTION
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C
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C CQRDC uses Householder transformations to compute the QR
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C factorization of an N by P matrix X. Column pivoting
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C based on the 2-norms of the reduced columns may be
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C performed at the users option.
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C
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C On Entry
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C
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C X COMPLEX(LDX,P), where LDX .GE. N.
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C X contains the matrix whose decomposition is to be
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C computed.
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C
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C LDX INTEGER.
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C LDX is the leading dimension of the array X.
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C
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C N INTEGER.
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C N is the number of rows of the matrix X.
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C
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C P INTEGER.
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C P is the number of columns of the matrix X.
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C
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C JVPT INTEGER(P).
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C JVPT contains integers that control the selection
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C of the pivot columns. The K-th column X(K) of X
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C is placed in one of three classes according to the
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C value of JVPT(K).
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C
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C If JVPT(K) .GT. 0, then X(K) is an initial
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C column.
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C
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C If JVPT(K) .EQ. 0, then X(K) is a free column.
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C
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C If JVPT(K) .LT. 0, then X(K) is a final column.
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C
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C Before the decomposition is computed, initial columns
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C are moved to the beginning of the array X and final
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C columns to the end. Both initial and final columns
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C are frozen in place during the computation and only
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C free columns are moved. At the K-th stage of the
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C reduction, if X(K) is occupied by a free column
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C it is interchanged with the free column of largest
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C reduced norm. JVPT is not referenced if
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C JOB .EQ. 0.
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C
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C WORK COMPLEX(P).
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C WORK is a work array. WORK is not referenced if
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C JOB .EQ. 0.
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C
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C JOB INTEGER.
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C JOB is an integer that initiates column pivoting.
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C If JOB .EQ. 0, no pivoting is done.
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C If JOB .NE. 0, pivoting is done.
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C
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C On Return
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C
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C X X contains in its upper triangle the upper
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C triangular matrix R of the QR factorization.
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C Below its diagonal X contains information from
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C which the unitary part of the decomposition
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C can be recovered. Note that if pivoting has
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C been requested, the decomposition is not that
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C of the original matrix X but that of X
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C with its columns permuted as described by JVPT.
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C
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C QRAUX COMPLEX(P).
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C QRAUX contains further information required to recover
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C the unitary part of the decomposition.
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C
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C JVPT JVPT(K) contains the index of the column of the
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C original matrix that has been interchanged into
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C the K-th column, if pivoting was requested.
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C
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C***REFERENCES J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W.
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C Stewart, LINPACK Users' Guide, SIAM, 1979.
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C***ROUTINES CALLED CAXPY, CDOTC, CSCAL, CSWAP, SCNRM2
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C***REVISION HISTORY (YYMMDD)
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C 780814 DATE WRITTEN
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C 890531 Changed all specific intrinsics to generic. (WRB)
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C 890831 Modified array declarations. (WRB)
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C 890831 REVISION DATE from Version 3.2
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C 891214 Prologue converted to Version 4.0 format. (BAB)
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C 900326 Removed duplicate information from DESCRIPTION section.
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C (WRB)
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C 920501 Reformatted the REFERENCES section. (WRB)
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C***END PROLOGUE CQRDC
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INTEGER LDX,N,P,JOB
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INTEGER JPVT(*)
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COMPLEX X(LDX,*),QRAUX(*),WORK(*)
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C
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INTEGER J,JP,L,LP1,LUP,MAXJ,PL,PU
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REAL MAXNRM,SCNRM2,TT
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COMPLEX CDOTC,NRMXL,T
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LOGICAL NEGJ,SWAPJ
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COMPLEX CSIGN,ZDUM,ZDUM1,ZDUM2
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REAL CABS1
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CSIGN(ZDUM1,ZDUM2) = ABS(ZDUM1)*(ZDUM2/ABS(ZDUM2))
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CABS1(ZDUM) = ABS(REAL(ZDUM)) + ABS(AIMAG(ZDUM))
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C
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C***FIRST EXECUTABLE STATEMENT CQRDC
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PL = 1
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PU = 0
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IF (JOB .EQ. 0) GO TO 60
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C
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C PIVOTING HAS BEEN REQUESTED. REARRANGE THE COLUMNS
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C ACCORDING TO JPVT.
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C
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DO 20 J = 1, P
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SWAPJ = JPVT(J) .GT. 0
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NEGJ = JPVT(J) .LT. 0
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JPVT(J) = J
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IF (NEGJ) JPVT(J) = -J
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IF (.NOT.SWAPJ) GO TO 10
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IF (J .NE. PL) CALL CSWAP(N,X(1,PL),1,X(1,J),1)
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JPVT(J) = JPVT(PL)
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JPVT(PL) = J
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PL = PL + 1
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10 CONTINUE
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20 CONTINUE
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PU = P
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DO 50 JJ = 1, P
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J = P - JJ + 1
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IF (JPVT(J) .GE. 0) GO TO 40
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JPVT(J) = -JPVT(J)
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IF (J .EQ. PU) GO TO 30
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CALL CSWAP(N,X(1,PU),1,X(1,J),1)
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JP = JPVT(PU)
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JPVT(PU) = JPVT(J)
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JPVT(J) = JP
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30 CONTINUE
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PU = PU - 1
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40 CONTINUE
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50 CONTINUE
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60 CONTINUE
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C
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C COMPUTE THE NORMS OF THE FREE COLUMNS.
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C
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IF (PU .LT. PL) GO TO 80
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DO 70 J = PL, PU
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QRAUX(J) = CMPLX(SCNRM2(N,X(1,J),1),0.0E0)
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WORK(J) = QRAUX(J)
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70 CONTINUE
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80 CONTINUE
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C
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C PERFORM THE HOUSEHOLDER REDUCTION OF X.
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C
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LUP = MIN(N,P)
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DO 200 L = 1, LUP
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IF (L .LT. PL .OR. L .GE. PU) GO TO 120
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C
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C LOCATE THE COLUMN OF LARGEST NORM AND BRING IT
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C INTO THE PIVOT POSITION.
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C
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MAXNRM = 0.0E0
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MAXJ = L
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DO 100 J = L, PU
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IF (REAL(QRAUX(J)) .LE. MAXNRM) GO TO 90
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MAXNRM = REAL(QRAUX(J))
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MAXJ = J
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90 CONTINUE
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100 CONTINUE
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IF (MAXJ .EQ. L) GO TO 110
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CALL CSWAP(N,X(1,L),1,X(1,MAXJ),1)
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QRAUX(MAXJ) = QRAUX(L)
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WORK(MAXJ) = WORK(L)
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JP = JPVT(MAXJ)
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JPVT(MAXJ) = JPVT(L)
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JPVT(L) = JP
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110 CONTINUE
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120 CONTINUE
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QRAUX(L) = (0.0E0,0.0E0)
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IF (L .EQ. N) GO TO 190
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C
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C COMPUTE THE HOUSEHOLDER TRANSFORMATION FOR COLUMN L.
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C
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NRMXL = CMPLX(SCNRM2(N-L+1,X(L,L),1),0.0E0)
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IF (CABS1(NRMXL) .EQ. 0.0E0) GO TO 180
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IF (CABS1(X(L,L)) .NE. 0.0E0)
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1 NRMXL = CSIGN(NRMXL,X(L,L))
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CALL CSCAL(N-L+1,(1.0E0,0.0E0)/NRMXL,X(L,L),1)
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X(L,L) = (1.0E0,0.0E0) + X(L,L)
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C
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C APPLY THE TRANSFORMATION TO THE REMAINING COLUMNS,
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C UPDATING THE NORMS.
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C
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LP1 = L + 1
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IF (P .LT. LP1) GO TO 170
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DO 160 J = LP1, P
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T = -CDOTC(N-L+1,X(L,L),1,X(L,J),1)/X(L,L)
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CALL CAXPY(N-L+1,T,X(L,L),1,X(L,J),1)
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IF (J .LT. PL .OR. J .GT. PU) GO TO 150
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IF (CABS1(QRAUX(J)) .EQ. 0.0E0) GO TO 150
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TT = 1.0E0 - (ABS(X(L,J))/REAL(QRAUX(J)))**2
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TT = MAX(TT,0.0E0)
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T = CMPLX(TT,0.0E0)
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TT = 1.0E0
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1 + 0.05E0*TT*(REAL(QRAUX(J))/REAL(WORK(J)))**2
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IF (TT .EQ. 1.0E0) GO TO 130
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QRAUX(J) = QRAUX(J)*SQRT(T)
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GO TO 140
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130 CONTINUE
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QRAUX(J) = CMPLX(SCNRM2(N-L,X(L+1,J),1),0.0E0)
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WORK(J) = QRAUX(J)
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140 CONTINUE
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150 CONTINUE
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160 CONTINUE
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170 CONTINUE
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C
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C SAVE THE TRANSFORMATION.
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C
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QRAUX(L) = X(L,L)
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X(L,L) = -NRMXL
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180 CONTINUE
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190 CONTINUE
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200 CONTINUE
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RETURN
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END
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