*> \brief \b SLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SLATRD + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW ) * * .. Scalar Arguments .. * CHARACTER UPLO * INTEGER LDA, LDW, N, NB * .. * .. Array Arguments .. * REAL A( LDA, * ), E( * ), TAU( * ), W( LDW, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SLATRD reduces NB rows and columns of a real symmetric matrix A to *> symmetric tridiagonal form by an orthogonal similarity *> transformation Q**T * A * Q, and returns the matrices V and W which are *> needed to apply the transformation to the unreduced part of A. *> *> If UPLO = 'U', SLATRD reduces the last NB rows and columns of a *> matrix, of which the upper triangle is supplied; *> if UPLO = 'L', SLATRD reduces the first NB rows and columns of a *> matrix, of which the lower triangle is supplied. *> *> This is an auxiliary routine called by SSYTRD. *> \endverbatim * * Arguments: * ========== * *> \param[in] UPLO *> \verbatim *> UPLO is CHARACTER*1 *> Specifies whether the upper or lower triangular part of the *> symmetric matrix A is stored: *> = 'U': Upper triangular *> = 'L': Lower triangular *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix A. *> \endverbatim *> *> \param[in] NB *> \verbatim *> NB is INTEGER *> The number of rows and columns to be reduced. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is REAL 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 last NB columns have been reduced to *> tridiagonal form, with the diagonal elements overwriting *> the diagonal elements of A; the elements above the diagonal *> with the array TAU, represent the orthogonal matrix Q as a *> product of elementary reflectors; *> if UPLO = 'L', the first NB columns have been reduced to *> tridiagonal form, with the diagonal elements overwriting *> the diagonal elements of A; the elements below the diagonal *> with the array TAU, represent the orthogonal matrix Q as a *> product of elementary reflectors. *> See Further Details. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= (1,N). *> \endverbatim *> *> \param[out] E *> \verbatim *> E is REAL array, dimension (N-1) *> If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal *> elements of the last NB columns of the reduced matrix; *> if UPLO = 'L', E(1:nb) contains the subdiagonal elements of *> the first NB columns of the reduced matrix. *> \endverbatim *> *> \param[out] TAU *> \verbatim *> TAU is REAL array, dimension (N-1) *> The scalar factors of the elementary reflectors, stored in *> TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'. *> See Further Details. *> \endverbatim *> *> \param[out] W *> \verbatim *> W is REAL array, dimension (LDW,NB) *> The n-by-nb matrix W required to update the unreduced part *> of A. *> \endverbatim *> *> \param[in] LDW *> \verbatim *> LDW is INTEGER *> The leading dimension of the array W. LDW >= max(1,N). *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup doubleOTHERauxiliary * *> \par Further Details: * ===================== *> *> \verbatim *> *> If UPLO = 'U', the matrix Q is represented as a product of elementary *> reflectors *> *> Q = H(n) H(n-1) . . . H(n-nb+1). *> *> Each H(i) has the form *> *> H(i) = I - tau * v * v**T *> *> where tau is a real scalar, and v is a real vector with *> v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), *> and tau in TAU(i-1). *> *> If UPLO = 'L', the matrix Q is represented as a product of elementary *> reflectors *> *> Q = H(1) H(2) . . . H(nb). *> *> Each H(i) has the form *> *> H(i) = I - tau * v * v**T *> *> where tau is a real scalar, and v is a real vector with *> v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), *> and tau in TAU(i). *> *> The elements of the vectors v together form the n-by-nb matrix V *> which is needed, with W, to apply the transformation to the unreduced *> part of the matrix, using a symmetric rank-2k update of the form: *> A := A - V*W**T - W*V**T. *> *> The contents of A on exit are illustrated by the following examples *> with n = 5 and nb = 2: *> *> if UPLO = 'U': if UPLO = 'L': *> *> ( a a a v4 v5 ) ( d ) *> ( a a v4 v5 ) ( 1 d ) *> ( a 1 v5 ) ( v1 1 a ) *> ( d 1 ) ( v1 v2 a a ) *> ( d ) ( v1 v2 a a a ) *> *> where d denotes a diagonal element of the reduced matrix, a denotes *> an element of the original matrix that is unchanged, and vi denotes *> an element of the vector defining H(i). *> \endverbatim *> * ===================================================================== SUBROUTINE SLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW ) * * -- LAPACK auxiliary 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 UPLO INTEGER LDA, LDW, N, NB * .. * .. Array Arguments .. REAL A( LDA, * ), E( * ), TAU( * ), W( LDW, * ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE, HALF PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, HALF = 0.5E+0 ) * .. * .. Local Scalars .. INTEGER I, IW REAL ALPHA * .. * .. External Subroutines .. EXTERNAL SAXPY, SGEMV, SLARFG, SSCAL, SSYMV * .. * .. External Functions .. LOGICAL LSAME REAL SDOT EXTERNAL LSAME, SDOT * .. * .. Intrinsic Functions .. INTRINSIC MIN * .. * .. Executable Statements .. * * Quick return if possible * IF( N.LE.0 ) $ RETURN * IF( LSAME( UPLO, 'U' ) ) THEN * * Reduce last NB columns of upper triangle * DO 10 I = N, N - NB + 1, -1 IW = I - N + NB IF( I.LT.N ) THEN * * Update A(1:i,i) * CALL SGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ), $ LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 ) CALL SGEMV( 'No transpose', I, N-I, -ONE, W( 1, IW+1 ), $ LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 ) END IF IF( I.GT.1 ) THEN * * Generate elementary reflector H(i) to annihilate * A(1:i-2,i) * CALL SLARFG( I-1, A( I-1, I ), A( 1, I ), 1, TAU( I-1 ) ) E( I-1 ) = A( I-1, I ) A( I-1, I ) = ONE * * Compute W(1:i-1,i) * CALL SSYMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1, $ ZERO, W( 1, IW ), 1 ) IF( I.LT.N ) THEN CALL SGEMV( 'Transpose', I-1, N-I, ONE, W( 1, IW+1 ), $ LDW, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 ) CALL SGEMV( 'No transpose', I-1, N-I, -ONE, $ A( 1, I+1 ), LDA, W( I+1, IW ), 1, ONE, $ W( 1, IW ), 1 ) CALL SGEMV( 'Transpose', I-1, N-I, ONE, A( 1, I+1 ), $ LDA, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 ) CALL SGEMV( 'No transpose', I-1, N-I, -ONE, $ W( 1, IW+1 ), LDW, W( I+1, IW ), 1, ONE, $ W( 1, IW ), 1 ) END IF CALL SSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 ) ALPHA = -HALF*TAU( I-1 )*SDOT( I-1, W( 1, IW ), 1, $ A( 1, I ), 1 ) CALL SAXPY( I-1, ALPHA, A( 1, I ), 1, W( 1, IW ), 1 ) END IF * 10 CONTINUE ELSE * * Reduce first NB columns of lower triangle * DO 20 I = 1, NB * * Update A(i:n,i) * CALL SGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ), $ LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 ) CALL SGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ), $ LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 ) IF( I.LT.N ) THEN * * Generate elementary reflector H(i) to annihilate * A(i+2:n,i) * CALL SLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1, $ TAU( I ) ) E( I ) = A( I+1, I ) A( I+1, I ) = ONE * * Compute W(i+1:n,i) * CALL SSYMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA, $ A( I+1, I ), 1, ZERO, W( I+1, I ), 1 ) CALL SGEMV( 'Transpose', N-I, I-1, ONE, W( I+1, 1 ), LDW, $ A( I+1, I ), 1, ZERO, W( 1, I ), 1 ) CALL SGEMV( 'No transpose', N-I, I-1, -ONE, A( I+1, 1 ), $ LDA, W( 1, I ), 1, ONE, W( I+1, I ), 1 ) CALL SGEMV( 'Transpose', N-I, I-1, ONE, A( I+1, 1 ), LDA, $ A( I+1, I ), 1, ZERO, W( 1, I ), 1 ) CALL SGEMV( 'No transpose', N-I, I-1, -ONE, W( I+1, 1 ), $ LDW, W( 1, I ), 1, ONE, W( I+1, I ), 1 ) CALL SSCAL( N-I, TAU( I ), W( I+1, I ), 1 ) ALPHA = -HALF*TAU( I )*SDOT( N-I, W( I+1, I ), 1, $ A( I+1, I ), 1 ) CALL SAXPY( N-I, ALPHA, A( I+1, I ), 1, W( I+1, I ), 1 ) END IF * 20 CONTINUE END IF * RETURN * * End of SLATRD * END