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OpenBLAS/lapack-netlib/SRC/dsptrd.f

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*> \brief \b DSPTRD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSPTRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsptrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsptrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsptrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AP( * ), D( * ), E( * ), TAU( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSPTRD reduces a real symmetric matrix A stored in packed form to
*> symmetric tridiagonal form T by an orthogonal similarity
*> transformation: Q**T * A * Q = T.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangle of the symmetric matrix
*> A, packed columnwise in a linear array. The j-th column of A
*> is stored in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*> 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.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The diagonal elements of the tridiagonal matrix T:
*> D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is 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'.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION 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 doubleOTHERcomputational
*
*> \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**T
*>
*> 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 AP,
*> overwriting A(1:i-1,i+1), and tau is stored 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**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+2:n) is stored on exit in AP,
*> overwriting A(i+2:n,i), and tau is stored in TAU(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DSPTRD( UPLO, N, AP, 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, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( * ), D( * ), E( * ), TAU( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO, HALF
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0,
$ HALF = 1.0D0 / 2.0D0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, I1, I1I1, II
DOUBLE PRECISION ALPHA, TAUI
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DLARFG, DSPMV, DSPR2, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DDOT
EXTERNAL LSAME, DDOT
* ..
* .. 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
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPTRD', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Reduce the upper triangle of A.
* I1 is the index in AP of A(1,I+1).
*
I1 = N*( N-1 ) / 2 + 1
DO 10 I = N - 1, 1, -1
*
* Generate elementary reflector H(i) = I - tau * v * v**T
* to annihilate A(1:i-1,i+1)
*
CALL DLARFG( I, AP( I1+I-1 ), AP( I1 ), 1, TAUI )
E( I ) = AP( I1+I-1 )
*
IF( TAUI.NE.ZERO ) THEN
*
* Apply H(i) from both sides to A(1:i,1:i)
*
AP( I1+I-1 ) = ONE
*
* Compute y := tau * A * v storing y in TAU(1:i)
*
CALL DSPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU,
$ 1 )
*
* Compute w := y - 1/2 * tau * (y**T *v) * v
*
ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, AP( I1 ), 1 )
CALL DAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 )
*
* Apply the transformation as a rank-2 update:
* A := A - v * w**T - w * v**T
*
CALL DSPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP )
*
AP( I1+I-1 ) = E( I )
END IF
D( I+1 ) = AP( I1+I )
TAU( I ) = TAUI
I1 = I1 - I
10 CONTINUE
D( 1 ) = AP( 1 )
ELSE
*
* Reduce the lower triangle of A. II is the index in AP of
* A(i,i) and I1I1 is the index of A(i+1,i+1).
*
II = 1
DO 20 I = 1, N - 1
I1I1 = II + N - I + 1
*
* Generate elementary reflector H(i) = I - tau * v * v**T
* to annihilate A(i+2:n,i)
*
CALL DLARFG( N-I, AP( II+1 ), AP( II+2 ), 1, TAUI )
E( I ) = AP( II+1 )
*
IF( TAUI.NE.ZERO ) THEN
*
* Apply H(i) from both sides to A(i+1:n,i+1:n)
*
AP( II+1 ) = ONE
*
* Compute y := tau * A * v storing y in TAU(i:n-1)
*
CALL DSPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ), 1,
$ ZERO, TAU( I ), 1 )
*
* Compute w := y - 1/2 * tau * (y**T *v) * v
*
ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, AP( II+1 ),
$ 1 )
CALL DAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 )
*
* Apply the transformation as a rank-2 update:
* A := A - v * w**T - w * v**T
*
CALL DSPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1,
$ AP( I1I1 ) )
*
AP( II+1 ) = E( I )
END IF
D( I ) = AP( II )
TAU( I ) = TAUI
II = I1I1
20 CONTINUE
D( N ) = AP( II )
END IF
*
RETURN
*
* End of DSPTRD
*
END
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