SUBROUTINE SLAPLL( N, X, INCX, Y, INCY, SSMIN ) * * -- LAPACK auxiliary routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. INTEGER INCX, INCY, N REAL SSMIN * .. * .. Array Arguments .. REAL X( * ), Y( * ) * .. * * Purpose * ======= * * Given two column vectors X and Y, let * * A = ( X Y ). * * The subroutine first computes the QR factorization of A = Q*R, * and then computes the SVD of the 2-by-2 upper triangular matrix R. * The smaller singular value of R is returned in SSMIN, which is used * as the measurement of the linear dependency of the vectors X and Y. * * Arguments * ========= * * N (input) INTEGER * The length of the vectors X and Y. * * X (input/output) REAL array, * dimension (1+(N-1)*INCX) * On entry, X contains the N-vector X. * On exit, X is overwritten. * * INCX (input) INTEGER * The increment between successive elements of X. INCX > 0. * * Y (input/output) REAL array, * dimension (1+(N-1)*INCY) * On entry, Y contains the N-vector Y. * On exit, Y is overwritten. * * INCY (input) INTEGER * The increment between successive elements of Y. INCY > 0. * * SSMIN (output) REAL * The smallest singular value of the N-by-2 matrix A = ( X Y ). * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) * .. * .. Local Scalars .. REAL A11, A12, A22, C, SSMAX, TAU * .. * .. External Functions .. REAL SDOT EXTERNAL SDOT * .. * .. External Subroutines .. EXTERNAL SAXPY, SLARFG, SLAS2 * .. * .. Executable Statements .. * * Quick return if possible * IF( N.LE.1 ) THEN SSMIN = ZERO RETURN END IF * * Compute the QR factorization of the N-by-2 matrix ( X Y ) * CALL SLARFG( N, X( 1 ), X( 1+INCX ), INCX, TAU ) A11 = X( 1 ) X( 1 ) = ONE * C = -TAU*SDOT( N, X, INCX, Y, INCY ) CALL SAXPY( N, C, X, INCX, Y, INCY ) * CALL SLARFG( N-1, Y( 1+INCY ), Y( 1+2*INCY ), INCY, TAU ) * A12 = Y( 1 ) A22 = Y( 1+INCY ) * * Compute the SVD of 2-by-2 Upper triangular matrix. * CALL SLAS2( A11, A12, A22, SSMIN, SSMAX ) * RETURN * * End of SLAPLL * END