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+*DECK ALST2F
+ SUBROUTINE ALST2F(MDIM,M,N,A,TAU)
+*
+*-----------------------------------------------------------------------
+*
+*Purpose:
+* to obtain the QR factorization of the matrix a using Householder
+* transformations. Use LAPACK's DGEQRF routine storage. Douple precision
+* routine.
+*
+*Copyright:
+* Copyright (C) 1993 Ecole Polytechnique de Montreal
+* This library is free software; you can redistribute it and/or
+* modify it under the terms of the GNU Lesser General Public
+* License as published by the Free Software Foundation; either
+* version 2.1 of the License, or (at your option) any later version
+*
+*Author(s): A. Hebert
+*
+*Reference:
+* P.A. BUSINGER, Num. Math. 7, 269-276 (1965).
+*
+*Parameters: input
+* MDIM dimensioned column length of A.
+* M number of rows of A
+* N number of columns of A. N.le.M is assumed.
+* A matrix A.
+*
+*Parameters: output
+* A decomposed matrix. On exit, the elements on and above the
+* diagonal of the array contain the m by n upper trapezoidal
+* matrix R (R is upper triangular if m >= n); the elements
+* below the diagonal, with the array TAU, represent the
+* orthogonal matrix Q as a product of elementary reflectors.
+* TAU scalar factors of the elementary reflectors.
+*
+*-----------------------------------------------------------------------
+*
+ IMPLICIT REAL(KIND=8)(A-H,O-Z)
+*----
+* SUBROUTINE ARGUMENTS
+*----
+ INTEGER MDIM,M,N
+ REAL(KIND=8) A(MDIM,N),TAU(N)
+*----
+* LOCAL VARIABLES
+*----
+ CHARACTER HSMG*131
+*----
+* ALLOCATABLE ARRAYS
+*----
+ REAL(KIND=8), ALLOCATABLE, DIMENSION(:,:) :: W
+*----
+* CHECK THE INPUT
+*----
+ IF(MDIM.LT.M) CALL XABORT('ALST2F: MDIM.LT.M')
+ IF(N.LT.1) CALL XABORT('ALST2F: N.LT.1')
+ IF(N.GT.M) THEN
+ WRITE(HSMG,'(18HALST2F: N.GT.M (N=,I3,3H M=,I3,2H).)') N,M
+ CALL XABORT(HSMG)
+ ENDIF
+*----
+* PERFORM QR FACTORIZATION.
+*----
+ ALLOCATE(W(M,1))
+ DO J=1,N
+ M1 = M-J+1; W(:M1,1) = A(J:M,J); X1 = W(1,1);
+ AX = SQRT(DOT_PRODUCT(W(:M1,1),W(:M1,1)))
+ A1 = ABS(X1); S = SIGN(1.0D0,W(1,1));
+ SSSS = -AX*S; A1 = A1+AX;
+ W(1,1) = A1*S
+ DD2 = A1*AX
+ IF(DD2 == 0.0D0) CALL XABORT('ALST2F: SINGULAR REFLECTION')
+ W(:M1,1) = W(:M1,1)/SQRT(DD2)
+ A(J:M,J) = W(:M1,1)
+ IF(J < N) THEN
+ A(J:M,J+1:N) = A(J:M,J+1:N)
+ 1 -MATMUL(W(:M1,:),(MATMUL(TRANSPOSE(W(:M1,:)),A(J:M,J+1:N))))
+ ENDIF
+ DIAG = A(J,J)
+ A(J:M,J) = A(J:M,J)/DIAG
+ A(J,J) = SSSS
+ TAU(J) = -DIAG*DIAG
+ ENDDO
+ DEALLOCATE(W)
+ RETURN
+ END