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REAL FUNCTION SQRT14( TRANS, M, N, NRHS, A, LDA, X,
$ LDX, WORK, LWORK ) * * -- LAPACK test routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. CHARACTER TRANS INTEGER LDA, LDX, LWORK, M, N, NRHS * .. * .. Array Arguments .. REAL A( LDA, * ), WORK( LWORK ), X( LDX, * ) * .. * * Purpose * ======= * * SQRT14 checks whether X is in the row space of A or A'. It does so * by scaling both X and A such that their norms are in the range * [sqrt(eps), 1/sqrt(eps)], then computing a QR factorization of [A,X] * (if TRANS = 'T') or an LQ factorization of [A',X]' (if TRANS = 'N'), * and returning the norm of the trailing triangle, scaled by * MAX(M,N,NRHS)*eps. * * Arguments * ========= * * TRANS (input) CHARACTER*1 * = 'N': No transpose, check for X in the row space of A * = 'T': Transpose, check for X in the row space of A'. * * M (input) INTEGER * The number of rows of the matrix A. * * N (input) INTEGER * The number of columns of the matrix A. * * NRHS (input) INTEGER * The number of right hand sides, i.e., the number of columns * of X. * * A (input) REAL array, dimension (LDA,N) * The M-by-N matrix A. * * LDA (input) INTEGER * The leading dimension of the array A. * * X (input) REAL array, dimension (LDX,NRHS) * If TRANS = 'N', the N-by-NRHS matrix X. * IF TRANS = 'T', the M-by-NRHS matrix X. * * LDX (input) INTEGER * The leading dimension of the array X. * * WORK (workspace) REAL array dimension (LWORK) * * LWORK (input) INTEGER * length of workspace array required * If TRANS = 'N', LWORK >= (M+NRHS)*(N+2); * if TRANS = 'T', LWORK >= (N+NRHS)*(M+2). * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 ) * .. * .. Local Scalars .. LOGICAL TPSD INTEGER I, INFO, J, LDWORK REAL ANRM, ERR, XNRM * .. * .. Local Arrays .. REAL RWORK( 1 ) * .. * .. External Functions .. LOGICAL LSAME REAL SLAMCH, SLANGE EXTERNAL LSAME, SLAMCH, SLANGE * .. * .. External Subroutines .. EXTERNAL SGELQ2, SGEQR2, SLACPY, SLASCL, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, REAL * .. * .. Executable Statements .. * SQRT14 = ZERO IF( LSAME( TRANS, 'N' ) ) THEN LDWORK = M + NRHS TPSD = .FALSE. IF( LWORK.LT.( M+NRHS )*( N+2 ) ) THEN CALL XERBLA( 'SQRT14', 10 ) RETURN ELSE IF( N.LE.0 .OR. NRHS.LE.0 ) THEN RETURN END IF ELSE IF( LSAME( TRANS, 'T' ) ) THEN LDWORK = M TPSD = .TRUE. IF( LWORK.LT.( N+NRHS )*( M+2 ) ) THEN CALL XERBLA( 'SQRT14', 10 ) RETURN ELSE IF( M.LE.0 .OR. NRHS.LE.0 ) THEN RETURN END IF ELSE CALL XERBLA( 'SQRT14', 1 ) RETURN END IF * * Copy and scale A * CALL SLACPY( 'All', M, N, A, LDA, WORK, LDWORK ) ANRM = SLANGE( 'M', M, N, WORK, LDWORK, RWORK ) IF( ANRM.NE.ZERO ) $ CALL SLASCL( 'G', 0, 0, ANRM, ONE, M, N, WORK, LDWORK, INFO ) * * Copy X or X' into the right place and scale it * IF( TPSD ) THEN * * Copy X into columns n+1:n+nrhs of work * CALL SLACPY( 'All', M, NRHS, X, LDX, WORK( N*LDWORK+1 ), $ LDWORK ) XNRM = SLANGE( 'M', M, NRHS, WORK( N*LDWORK+1 ), LDWORK, $ RWORK ) IF( XNRM.NE.ZERO ) $ CALL SLASCL( 'G', 0, 0, XNRM, ONE, M, NRHS, $ WORK( N*LDWORK+1 ), LDWORK, INFO ) ANRM = SLANGE( 'One-norm', M, N+NRHS, WORK, LDWORK, RWORK ) * * Compute QR factorization of X * CALL SGEQR2( M, N+NRHS, WORK, LDWORK, $ WORK( LDWORK*( N+NRHS )+1 ), $ WORK( LDWORK*( N+NRHS )+MIN( M, N+NRHS )+1 ), $ INFO ) * * Compute largest entry in upper triangle of * work(n+1:m,n+1:n+nrhs) * ERR = ZERO DO 20 J = N + 1, N + NRHS DO 10 I = N + 1, MIN( M, J ) ERR = MAX( ERR, ABS( WORK( I+( J-1 )*M ) ) ) 10 CONTINUE 20 CONTINUE * ELSE * * Copy X' into rows m+1:m+nrhs of work * DO 40 I = 1, N DO 30 J = 1, NRHS WORK( M+J+( I-1 )*LDWORK ) = X( I, J ) 30 CONTINUE 40 CONTINUE * XNRM = SLANGE( 'M', NRHS, N, WORK( M+1 ), LDWORK, RWORK ) IF( XNRM.NE.ZERO ) $ CALL SLASCL( 'G', 0, 0, XNRM, ONE, NRHS, N, WORK( M+1 ), $ LDWORK, INFO ) * * Compute LQ factorization of work * CALL SGELQ2( LDWORK, N, WORK, LDWORK, WORK( LDWORK*N+1 ), $ WORK( LDWORK*( N+1 )+1 ), INFO ) * * Compute largest entry in lower triangle in * work(m+1:m+nrhs,m+1:n) * ERR = ZERO DO 60 J = M + 1, N DO 50 I = J, LDWORK ERR = MAX( ERR, ABS( WORK( I+( J-1 )*LDWORK ) ) ) 50 CONTINUE 60 CONTINUE * END IF * SQRT14 = ERR / ( REAL( MAX( M, N, NRHS ) )*SLAMCH( 'Epsilon' ) ) * RETURN * * End of SQRT14 * END |