*> \brief \b SORT01 * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * * Definition: * =========== * * SUBROUTINE SORT01( ROWCOL, M, N, U, LDU, WORK, LWORK, RESID ) * * .. Scalar Arguments .. * CHARACTER ROWCOL * INTEGER LDU, LWORK, M, N * REAL RESID * .. * .. Array Arguments .. * REAL U( LDU, * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SORT01 checks that the matrix U is orthogonal by computing the ratio *> *> RESID = norm( I - U*U' ) / ( n * EPS ), if ROWCOL = 'R', *> or *> RESID = norm( I - U'*U ) / ( m * EPS ), if ROWCOL = 'C'. *> *> Alternatively, if there isn't sufficient workspace to form *> I - U*U' or I - U'*U, the ratio is computed as *> *> RESID = abs( I - U*U' ) / ( n * EPS ), if ROWCOL = 'R', *> or *> RESID = abs( I - U'*U ) / ( m * EPS ), if ROWCOL = 'C'. *> *> where EPS is the machine precision. ROWCOL is used only if m = n; *> if m > n, ROWCOL is assumed to be 'C', and if m < n, ROWCOL is *> assumed to be 'R'. *> \endverbatim * * Arguments: * ========== * *> \param[in] ROWCOL *> \verbatim *> ROWCOL is CHARACTER *> Specifies whether the rows or columns of U should be checked *> for orthogonality. Used only if M = N. *> = 'R': Check for orthogonal rows of U *> = 'C': Check for orthogonal columns of U *> \endverbatim *> *> \param[in] M *> \verbatim *> M is INTEGER *> The number of rows of the matrix U. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The number of columns of the matrix U. *> \endverbatim *> *> \param[in] U *> \verbatim *> U is REAL array, dimension (LDU,N) *> The orthogonal matrix U. U is checked for orthogonal columns *> if m > n or if m = n and ROWCOL = 'C'. U is checked for *> orthogonal rows if m < n or if m = n and ROWCOL = 'R'. *> \endverbatim *> *> \param[in] LDU *> \verbatim *> LDU is INTEGER *> The leading dimension of the array U. LDU >= max(1,M). *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is REAL array, dimension (LWORK) *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> The length of the array WORK. For best performance, LWORK *> should be at least N*(N+1) if ROWCOL = 'C' or M*(M+1) if *> ROWCOL = 'R', but the test will be done even if LWORK is 0. *> \endverbatim *> *> \param[out] RESID *> \verbatim *> RESID is REAL *> RESID = norm( I - U * U' ) / ( n * EPS ), if ROWCOL = 'R', or *> RESID = norm( I - U' * U ) / ( m * EPS ), if ROWCOL = 'C'. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup single_eig * * ===================================================================== SUBROUTINE SORT01( ROWCOL, M, N, U, LDU, WORK, LWORK, RESID ) * * -- LAPACK test routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. CHARACTER ROWCOL INTEGER LDU, LWORK, M, N REAL RESID * .. * .. Array Arguments .. REAL U( LDU, * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) * .. * .. Local Scalars .. CHARACTER TRANSU INTEGER I, J, K, LDWORK, MNMIN REAL EPS, TMP * .. * .. External Functions .. LOGICAL LSAME REAL SDOT, SLAMCH, SLANSY EXTERNAL LSAME, SDOT, SLAMCH, SLANSY * .. * .. External Subroutines .. EXTERNAL SLASET, SSYRK * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN, REAL * .. * .. Executable Statements .. * RESID = ZERO * * Quick return if possible * IF( M.LE.0 .OR. N.LE.0 ) $ RETURN * EPS = SLAMCH( 'Precision' ) IF( M.LT.N .OR. ( M.EQ.N .AND. LSAME( ROWCOL, 'R' ) ) ) THEN TRANSU = 'N' K = N ELSE TRANSU = 'T' K = M END IF MNMIN = MIN( M, N ) * IF( ( MNMIN+1 )*MNMIN.LE.LWORK ) THEN LDWORK = MNMIN ELSE LDWORK = 0 END IF IF( LDWORK.GT.0 ) THEN * * Compute I - U*U' or I - U'*U. * CALL SLASET( 'Upper', MNMIN, MNMIN, ZERO, ONE, WORK, LDWORK ) CALL SSYRK( 'Upper', TRANSU, MNMIN, K, -ONE, U, LDU, ONE, WORK, $ LDWORK ) * * Compute norm( I - U*U' ) / ( K * EPS ) . * RESID = SLANSY( '1', 'Upper', MNMIN, WORK, LDWORK, $ WORK( LDWORK*MNMIN+1 ) ) RESID = ( RESID / REAL( K ) ) / EPS ELSE IF( TRANSU.EQ.'T' ) THEN * * Find the maximum element in abs( I - U'*U ) / ( m * EPS ) * DO 20 J = 1, N DO 10 I = 1, J IF( I.NE.J ) THEN TMP = ZERO ELSE TMP = ONE END IF TMP = TMP - SDOT( M, U( 1, I ), 1, U( 1, J ), 1 ) RESID = MAX( RESID, ABS( TMP ) ) 10 CONTINUE 20 CONTINUE RESID = ( RESID / REAL( M ) ) / EPS ELSE * * Find the maximum element in abs( I - U*U' ) / ( n * EPS ) * DO 40 J = 1, M DO 30 I = 1, J IF( I.NE.J ) THEN TMP = ZERO ELSE TMP = ONE END IF TMP = TMP - SDOT( N, U( J, 1 ), LDU, U( I, 1 ), LDU ) RESID = MAX( RESID, ABS( TMP ) ) 30 CONTINUE 40 CONTINUE RESID = ( RESID / REAL( N ) ) / EPS END IF RETURN * * End of SORT01 * END