sqrt16.f

Go to the documentation of this file.

      SUBROUTINE sqrt16( TRANS, M, N, NRHS, A, LDA, X, LDX, B, LDB,
     $                   rwork, resid )
*
*  -- LAPACK test routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
      CHARACTER          trans
      INTEGER            lda, ldb, ldx, m, n, nrhs
      REAL               resid
*     ..
*     .. Array Arguments ..
      REAL               a( lda, * ), b( ldb, * ), rwork( * ),
     $                   x( ldx, * )
*     ..
*
*  Purpose
*  =======
*
*  SQRT16 computes the residual for a solution of a system of linear
*  equations  A*x = b  or  A'*x = b:
*     RESID = norm(B - A*X) / ( max(m,n) * (norm(A) * norm(X)+ norm(NRHS)) * EPS ),
*  where EPS is the machine epsilon.
*
*  Arguments
*  =========
*
*  TRANS   (input) CHARACTER*1
*          Specifies the form of the system of equations:
*          = 'N':  A *x = b
*          = 'T':  A'*x = b, where A' is the transpose of A
*          = 'C':  A'*x = b, where A' is the transpose of A
*
*  M       (input) INTEGER
*          The number of rows of the matrix A.  M >= 0.
*
*  N       (input) INTEGER
*          The number of columns of the matrix A.  N >= 0.
*
*  NRHS    (input) INTEGER
*          The number of columns of B, the matrix of right hand sides.
*          NRHS >= 0.
*
*  A       (input) REAL array, dimension (LDA,N)
*          The original M x N matrix A.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,M).
*
*  X       (input) REAL array, dimension (LDX,NRHS)
*          The computed solution vectors for the system of linear
*          equations.
*
*  LDX     (input) INTEGER
*          The leading dimension of the array X.  If TRANS = 'N',
*          LDX >= max(1,N); if TRANS = 'T' or 'C', LDX >= max(1,M).
*
*  B       (input/output) REAL array, dimension (LDB,NRHS)
*          On entry, the right hand side vectors for the system of
*          linear equations.
*          On exit, B is overwritten with the difference B - A*X.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B.  IF TRANS = 'N',
*          LDB >= max(1,M); if TRANS = 'T' or 'C', LDB >= max(1,N).
*
*  RWORK   (workspace) REAL array, dimension (M)
*
*  RESID   (output) REAL
*          The maximum over the number of right hand sides of
*          norm(B - A*X) / ( max(m,n) * norm(A) * norm(X) * EPS ).
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               zero, one
      parameter( zero = 0.0e+0, one = 1.0e+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            j, n1, n2
      REAL               anorm, bnorm, eps, xnorm, rhsnorm
*     ..
*     .. External Functions ..
      LOGICAL            lsame
      REAL               sasum, slamch, slange
      EXTERNAL           lsame, sasum, slamch, slange
*     ..
*     .. External Subroutines ..
      EXTERNAL           sgemm
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max
*     ..
*     .. Executable Statements ..
*
*     Quick exit if M = 0 or N = 0 or NRHS = 0
*
      IF( m.LE.0 .OR. n.LE.0 .OR. nrhs.EQ.0 ) THEN
         resid = zero
         return
      END IF
*
      IF( lsame( trans, 'T' ) .OR. lsame( trans, 'C' ) ) THEN
         anorm = slange( 'I', m, n, a, lda, rwork )
         n1 = n
         n2 = m
      ELSE
         anorm = slange( '1', m, n, a, lda, rwork )
         n1 = m
         n2 = n
      END IF
*
      rhsnorm = slange( 'I', n, nrhs, b, ldb, rwork )
      eps = slamch( 'Epsilon' )
*
*     Compute  B - A*X  (or  B - A'*X ) and store in B.
*
      CALL sgemm( trans, 'No transpose', n1, nrhs, n2, -one, a, lda, x,
     $            ldx, one, b, ldb )
*
*     Compute the maximum over the number of right hand sides of
*        norm(B - A*X) / ( max(m,n) * norm(A) * norm(X) * EPS ) .
*
      resid = zero
      DO 10 j = 1, nrhs
         bnorm = sasum( n1, b( 1, j ), 1 )
         xnorm = sasum( n2, x( 1, j ), 1 )
         IF( anorm.EQ.zero .AND. bnorm.EQ.zero ) THEN
            resid = zero
         ELSE IF( anorm.LE.zero .OR. xnorm.LE.zero ) THEN
            resid = one / eps
         ELSE
            resid = max( resid, (  bnorm) / (anorm * xnorm + rhsnorm ) *
     $              ( max( m, n )*eps ) )
         END IF
   10 continue
*
      return
*
*     End of SQRT16
*
      END