*DECK DQRFAC SUBROUTINE DQRFAC (M, N, A, LDA, PIVOT, IPVT, LIPVT, SIGMA, + ACNORM, WA) C***BEGIN PROLOGUE DQRFAC C***SUBSIDIARY C***PURPOSE Subsidiary to DNLS1, DNLS1E, DNSQ and DNSQE C***LIBRARY SLATEC C***TYPE DOUBLE PRECISION (QRFAC-S, DQRFAC-D) C***AUTHOR (UNKNOWN) C***DESCRIPTION C C **** Double Precision version of QRFAC **** C C This subroutine uses Householder transformations with column C pivoting (optional) to compute a QR factorization of the C M by N matrix A. That is, DQRFAC determines an orthogonal C matrix Q, a permutation matrix P, and an upper trapezoidal C matrix R with diagonal elements of nonincreasing magnitude, C such that A*P = Q*R. The Householder transformation for C column K, K = 1,2,...,MIN(M,N), is of the form C C T C I - (1/U(K))*U*U C C where U has zeros in the first K-1 positions. The form of C this transformation and the method of pivoting first C appeared in the corresponding LINPACK subroutine. C C The subroutine statement is C C SUBROUTINE DQRFAC(M,N,A,LDA,PIVOT,IPVT,LIPVT,SIGMA,ACNORM,WA) C C where C C M is a positive integer input variable set to the number C of rows of A. C C N is a positive integer input variable set to the number C of columns of A. C C A is an M by N array. On input A contains the matrix for C which the QR factorization is to be computed. On output C the strict upper trapezoidal part of A contains the strict C upper trapezoidal part of R, and the lower trapezoidal C part of A contains a factored form of Q (the non-trivial C elements of the U vectors described above). C C LDA is a positive integer input variable not less than M C which specifies the leading dimension of the array A. C C PIVOT is a logical input variable. If pivot is set .TRUE., C then column pivoting is enforced. If pivot is set .FALSE., C then no column pivoting is done. C C IPVT is an integer output array of length LIPVT. IPVT C defines the permutation matrix P such that A*P = Q*R. C Column J of P is column IPVT(J) of the identity matrix. C If pivot is .FALSE., IPVT is not referenced. C C LIPVT is a positive integer input variable. If PIVOT is C .FALSE., then LIPVT may be as small as 1. If PIVOT is C .TRUE., then LIPVT must be at least N. C C SIGMA is an output array of length N which contains the C diagonal elements of R. C C ACNORM is an output array of length N which contains the C norms of the corresponding columns of the input matrix A. C If this information is not needed, then ACNORM can coincide C with SIGMA. C C WA is a work array of length N. If pivot is .FALSE., then WA C can coincide with SIGMA. C C***SEE ALSO DNLS1, DNLS1E, DNSQ, DNSQE C***ROUTINES CALLED D1MACH, DENORM C***REVISION HISTORY (YYMMDD) C 800301 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 890831 Modified array declarations. (WRB) C 891214 Prologue converted to Version 4.0 format. (BAB) C 900326 Removed duplicate information from DESCRIPTION section. C (WRB) C 900328 Added TYPE section. (WRB) C***END PROLOGUE DQRFAC