*> \brief \b CHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm). * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download CHETD2 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE CHETD2( UPLO, N, A, LDA, D, E, TAU, INFO ) * * .. Scalar Arguments .. * CHARACTER UPLO * INTEGER INFO, LDA, N * .. * .. Array Arguments .. * REAL D( * ), E( * ) * COMPLEX A( LDA, * ), TAU( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> CHETD2 reduces a complex Hermitian matrix A to real symmetric *> tridiagonal form T by a unitary similarity transformation: *> Q**H * A * Q = T. *> \endverbatim * * Arguments: * ========== * *> \param[in] UPLO *> \verbatim *> UPLO is CHARACTER*1 *> Specifies whether the upper or lower triangular part of the *> Hermitian matrix A is stored: *> = 'U': Upper triangular *> = 'L': Lower triangular *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix A. N >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is COMPLEX array, dimension (LDA,N) *> On entry, the Hermitian 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 unitary *> 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 unitary 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 >= max(1,N). *> \endverbatim *> *> \param[out] D *> \verbatim *> D is REAL array, dimension (N) *> The diagonal elements of the tridiagonal matrix T: *> D(i) = A(i,i). *> \endverbatim *> *> \param[out] E *> \verbatim *> E is REAL 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'. *> \endverbatim *> *> \param[out] TAU *> \verbatim *> TAU is COMPLEX array, dimension (N-1) *> The scalar factors of the elementary reflectors (see Further *> Details). *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> < 0: if INFO = -i, the i-th argument had an illegal value. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup complexHEcomputational * *> \par Further Details: * ===================== *> *> \verbatim *> *> 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**H *> *> where tau is a complex scalar, and v is a complex 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**H *> *> where tau is a complex scalar, and v is a complex 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). *> \endverbatim *> * ===================================================================== SUBROUTINE CHETD2( UPLO, N, A, LDA, D, E, TAU, INFO ) * * -- LAPACK computational routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * December 2016 * * .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, N * .. * .. Array Arguments .. REAL D( * ), E( * ) COMPLEX A( LDA, * ), TAU( * ) * .. * * ===================================================================== * * .. Parameters .. COMPLEX ONE, ZERO, HALF PARAMETER ( ONE = ( 1.0E+0, 0.0E+0 ), \$ ZERO = ( 0.0E+0, 0.0E+0 ), \$ HALF = ( 0.5E+0, 0.0E+0 ) ) * .. * .. Local Scalars .. LOGICAL UPPER INTEGER I COMPLEX ALPHA, TAUI * .. * .. External Subroutines .. EXTERNAL CAXPY, CHEMV, CHER2, CLARFG, XERBLA * .. * .. External Functions .. LOGICAL LSAME COMPLEX CDOTC EXTERNAL LSAME, CDOTC * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN, REAL * .. * .. 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( 'CHETD2', -INFO ) RETURN END IF * * Quick return if possible * IF( N.LE.0 ) \$ RETURN * IF( UPPER ) THEN * * Reduce the upper triangle of A * A( N, N ) = REAL( A( N, N ) ) DO 10 I = N - 1, 1, -1 * * Generate elementary reflector H(i) = I - tau * v * v**H * to annihilate A(1:i-1,i+1) * ALPHA = A( I, I+1 ) CALL CLARFG( I, ALPHA, A( 1, I+1 ), 1, TAUI ) E( I ) = ALPHA * 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 CHEMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1, ZERO, \$ TAU, 1 ) * * Compute w := x - 1/2 * tau * (x**H * v) * v * ALPHA = -HALF*TAUI*CDOTC( I, TAU, 1, A( 1, I+1 ), 1 ) CALL CAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 ) * * Apply the transformation as a rank-2 update: * A := A - v * w**H - w * v**H * CALL CHER2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A, \$ LDA ) * ELSE A( I, I ) = REAL( A( I, I ) ) END IF A( I, I+1 ) = E( I ) 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 * A( 1, 1 ) = REAL( A( 1, 1 ) ) DO 20 I = 1, N - 1 * * Generate elementary reflector H(i) = I - tau * v * v**H * to annihilate A(i+2:n,i) * ALPHA = A( I+1, I ) CALL CLARFG( N-I, ALPHA, A( MIN( I+2, N ), I ), 1, TAUI ) E( I ) = ALPHA * 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 CHEMV( 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**H * v) * v * ALPHA = -HALF*TAUI*CDOTC( N-I, TAU( I ), 1, A( I+1, I ), \$ 1 ) CALL CAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 ) * * Apply the transformation as a rank-2 update: * A := A - v * w**H - w * v**H * CALL CHER2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1, \$ A( I+1, I+1 ), LDA ) * ELSE A( I+1, I+1 ) = REAL( A( I+1, I+1 ) ) END IF A( I+1, I ) = E( I ) D( I ) = A( I, I ) TAU( I ) = TAUI 20 CONTINUE D( N ) = A( N, N ) END IF * RETURN * * End of CHETD2 * END