*DECK DHELS SUBROUTINE DHELS (A, LDA, N, Q, B) C***BEGIN PROLOGUE DHELS C***SUBSIDIARY C***PURPOSE Internal routine for DGMRES. C***LIBRARY SLATEC (SLAP) C***CATEGORY D2A4, D2B4 C***TYPE DOUBLE PRECISION (SHELS-S, DHELS-D) C***KEYWORDS GENERALIZED MINIMUM RESIDUAL, ITERATIVE PRECONDITION, C NON-SYMMETRIC LINEAR SYSTEM, SLAP, SPARSE C***AUTHOR Brown, Peter, (LLNL), pnbrown@llnl.gov C Hindmarsh, Alan, (LLNL), alanh@llnl.gov C Seager, Mark K., (LLNL), seager@llnl.gov C Lawrence Livermore National Laboratory C PO Box 808, L-60 C Livermore, CA 94550 (510) 423-3141 C***DESCRIPTION C This routine is extracted from the LINPACK routine SGESL with C changes due to the fact that A is an upper Hessenberg matrix. C C DHELS solves the least squares problem: C C MIN(B-A*X,B-A*X) C C using the factors computed by DHEQR. C C *Usage: C INTEGER LDA, N C DOUBLE PRECISION A(LDA,N), Q(2*N), B(N+1) C C CALL DHELS(A, LDA, N, Q, B) C C *Arguments: C A :IN Double Precision A(LDA,N) C The output from DHEQR which contains the upper C triangular factor R in the QR decomposition of A. C LDA :IN Integer C The leading dimension of the array A. C N :IN Integer C A is originally an (N+1) by N matrix. C Q :IN Double Precision Q(2*N) C The coefficients of the N Givens rotations C used in the QR factorization of A. C B :INOUT Double Precision B(N+1) C On input, B is the right hand side vector. C On output, B is the solution vector X. C C***SEE ALSO DGMRES C***ROUTINES CALLED DAXPY C***REVISION HISTORY (YYMMDD) C 890404 DATE WRITTEN C 890404 Previous REVISION DATE C 890915 Made changes requested at July 1989 CML Meeting. (MKS) C 890922 Numerous changes to prologue to make closer to SLATEC C standard. (FNF) C 890929 Numerous changes to reduce SP/DP differences. (FNF) C 910411 Prologue converted to Version 4.0 format. (BAB) C 910502 Added C***FIRST EXECUTABLE STATEMENT line. (FNF) C 910506 Made subsidiary to DGMRES. (FNF) C 920511 Added complete declaration section. (WRB) C***END PROLOGUE DHELS C The following is for optimized compilation on LLNL/LTSS Crays. CLLL. OPTIMIZE C .. Scalar Arguments .. INTEGER LDA, N C .. Array Arguments .. DOUBLE PRECISION A(LDA,*), B(*), Q(*) C .. Local Scalars .. DOUBLE PRECISION C, S, T, T1, T2 INTEGER IQ, K, KB, KP1 C .. External Subroutines .. EXTERNAL DAXPY C***FIRST EXECUTABLE STATEMENT DHELS C C Minimize(B-A*X,B-A*X). First form Q*B. C DO 20 K = 1, N KP1 = K + 1 IQ = 2*(K-1) + 1 C = Q(IQ) S = Q(IQ+1) T1 = B(K) T2 = B(KP1) B(K) = C*T1 - S*T2 B(KP1) = S*T1 + C*T2 20 CONTINUE C C Now solve R*X = Q*B. C DO 40 KB = 1, N K = N + 1 - KB B(K) = B(K)/A(K,K) T = -B(K) CALL DAXPY(K-1, T, A(1,K), 1, B(1), 1) 40 CONTINUE RETURN C------------- LAST LINE OF DHELS FOLLOWS ---------------------------- END