SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO ) * * -- LAPACK routine (version 3.2) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, N * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ) * .. * * Purpose * ======= * * DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal * form T by an orthogonal similarity transformation: Q' * A * Q = T. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * Specifies whether the upper or lower triangular part of the * symmetric matrix A is stored: * = 'U': Upper triangular * = 'L': Lower triangular * * N (input) INTEGER * The order of the matrix A. N >= 0. * * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) * On entry, the symmetric matrix A. If UPLO = 'U', the leading * n-by-n upper triangular part of A contains the upper * triangular part of the matrix A, and the strictly lower * triangular part of A is not referenced. If UPLO = 'L', the * leading n-by-n lower triangular part of A contains the lower * triangular part of the matrix A, and the strictly upper * triangular part of A is not referenced. * On exit, if UPLO = 'U', the diagonal and first superdiagonal * of A are overwritten by the corresponding elements of the * tridiagonal matrix T, and the elements above the first * superdiagonal, with the array TAU, represent the orthogonal * matrix Q as a product of elementary reflectors; if UPLO * = 'L', the diagonal and first subdiagonal of A are over- * written by the corresponding elements of the tridiagonal * matrix T, and the elements below the first subdiagonal, with * the array TAU, represent the orthogonal matrix Q as a product * of elementary reflectors. See Further Details. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * D (output) DOUBLE PRECISION array, dimension (N) * The diagonal elements of the tridiagonal matrix T: * D(i) = A(i,i). * * E (output) DOUBLE PRECISION array, dimension (N-1) * The off-diagonal elements of the tridiagonal matrix T: * E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. * * TAU (output) DOUBLE PRECISION array, dimension (N-1) * The scalar factors of the elementary reflectors (see Further * Details). * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value. * * Further Details * =============== * * If UPLO = 'U', the matrix Q is represented as a product of elementary * reflectors * * Q = H(n-1) . . . H(2) H(1). * * Each H(i) has the form * * H(i) = I - tau * v * v' * * where tau is a real scalar, and v is a real vector with * v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in * A(1:i-1,i+1), and tau in TAU(i). * * If UPLO = 'L', the matrix Q is represented as a product of elementary * reflectors * * Q = H(1) H(2) . . . H(n-1). * * Each H(i) has the form * * H(i) = I - tau * v * v' * * where tau is a real scalar, and v is a real vector with * v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), * and tau in TAU(i). * * The contents of A on exit are illustrated by the following examples * with n = 5: * * if UPLO = 'U': if UPLO = 'L': * * ( d e v2 v3 v4 ) ( d ) * ( d e v3 v4 ) ( e d ) * ( d e v4 ) ( v1 e d ) * ( d e ) ( v1 v2 e d ) * ( d ) ( v1 v2 v3 e d ) * * where d and e denote diagonal and off-diagonal elements of T, and vi * denotes an element of the vector defining H(i). * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ONE, ZERO, HALF PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0, $ HALF = 1.0D0 / 2.0D0 ) * .. * .. Local Scalars .. LOGICAL UPPER INTEGER I DOUBLE PRECISION ALPHA, TAUI * .. * .. External Subroutines .. EXTERNAL DAXPY, DLARFG, DSYMV, DSYR2, XERBLA * .. * .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DDOT EXTERNAL LSAME, DDOT * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. Executable Statements .. * * Test the input parameters * INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYTD2', -INFO ) RETURN END IF * * Quick return if possible * IF( N.LE.0 ) $ RETURN * IF( UPPER ) THEN * * Reduce the upper triangle of A * DO 10 I = N - 1, 1, -1 * * Generate elementary reflector H(i) = I - tau * v * v' * to annihilate A(1:i-1,i+1) * CALL DLARFG( I, A( I, I+1 ), A( 1, I+1 ), 1, TAUI ) E( I ) = A( I, I+1 ) * IF( TAUI.NE.ZERO ) THEN * * Apply H(i) from both sides to A(1:i,1:i) * A( I, I+1 ) = ONE * * Compute x := tau * A * v storing x in TAU(1:i) * CALL DSYMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1, ZERO, $ TAU, 1 ) * * Compute w := x - 1/2 * tau * (x'*v) * v * ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, A( 1, I+1 ), 1 ) CALL DAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 ) * * Apply the transformation as a rank-2 update: * A := A - v * w' - w * v' * CALL DSYR2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A, $ LDA ) * A( I, I+1 ) = E( I ) END IF D( I+1 ) = A( I+1, I+1 ) TAU( I ) = TAUI 10 CONTINUE D( 1 ) = A( 1, 1 ) ELSE * * Reduce the lower triangle of A * DO 20 I = 1, N - 1 * * Generate elementary reflector H(i) = I - tau * v * v' * to annihilate A(i+2:n,i) * CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1, $ TAUI ) E( I ) = A( I+1, I ) * IF( TAUI.NE.ZERO ) THEN * * Apply H(i) from both sides to A(i+1:n,i+1:n) * A( I+1, I ) = ONE * * Compute x := tau * A * v storing y in TAU(i:n-1) * CALL DSYMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA, $ A( I+1, I ), 1, ZERO, TAU( I ), 1 ) * * Compute w := x - 1/2 * tau * (x'*v) * v * ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, A( I+1, I ), $ 1 ) CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 ) * * Apply the transformation as a rank-2 update: * A := A - v * w' - w * v' * CALL DSYR2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1, $ A( I+1, I+1 ), LDA ) * A( I+1, I ) = E( I ) END IF D( I ) = A( I, I ) TAU( I ) = TAUI 20 CONTINUE D( N ) = A( N, N ) END IF * RETURN * * End of DSYTD2 * END