SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( * ), D( * ), E( * ), TAU( * )
* ..
*
* Purpose
* =======
*
* 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.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* AP (input/output) 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.
*
* 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 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'
*
* 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).
*
* =====================================================================
*
* .. 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'
* 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'*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' - w * v'
*
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'
* 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'*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' - w * v'
*
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