spteqr()

subroutine spteqr	(	character	compz,
		integer	n,
		real, dimension( * )	d,
		real, dimension( * )	e,
		real, dimension( ldz, * )	z,
		integer	ldz,
		real, dimension( * )	work,
		integer	info
	)

SPTEQR

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Purpose:

 SPTEQR computes all eigenvalues and, optionally, eigenvectors of a
 symmetric positive definite tridiagonal matrix by first factoring the
 matrix using SPTTRF, and then calling SBDSQR to compute the singular
 values of the bidiagonal factor.

 This routine computes the eigenvalues of the positive definite
 tridiagonal matrix to high relative accuracy.  This means that if the
 eigenvalues range over many orders of magnitude in size, then the
 small eigenvalues and corresponding eigenvectors will be computed
 more accurately than, for example, with the standard QR method.

 The eigenvectors of a full or band symmetric positive definite matrix
 can also be found if SSYTRD, SSPTRD, or SSBTRD has been used to
 reduce this matrix to tridiagonal form. (The reduction to tridiagonal
 form, however, may preclude the possibility of obtaining high
 relative accuracy in the small eigenvalues of the original matrix, if
 these eigenvalues range over many orders of magnitude.)

Parameters

[in]	COMPZ	COMPZ is CHARACTER*1 = 'N': Compute eigenvalues only. = 'V': Compute eigenvectors of original symmetric matrix also. Array Z contains the orthogonal matrix used to reduce the original matrix to tridiagonal form. = 'I': Compute eigenvectors of tridiagonal matrix also.
[in]	N	N is INTEGER The order of the matrix. N >= 0.
[in,out]	D	D is REAL array, dimension (N) On entry, the n diagonal elements of the tridiagonal matrix. On normal exit, D contains the eigenvalues, in descending order.
[in,out]	E	E is REAL array, dimension (N-1) On entry, the (n-1) subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed.
[in,out]	Z	Z is REAL array, dimension (LDZ, N) On entry, if COMPZ = 'V', the orthogonal matrix used in the reduction to tridiagonal form. On exit, if COMPZ = 'V', the orthonormal eigenvectors of the original symmetric matrix; if COMPZ = 'I', the orthonormal eigenvectors of the tridiagonal matrix. If INFO > 0 on exit, Z contains the eigenvectors associated with only the stored eigenvalues. If COMPZ = 'N', then Z is not referenced.
[in]	LDZ	LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if COMPZ = 'V' or 'I', LDZ >= max(1,N).
[out]	WORK	WORK is REAL array, dimension (4*N)
[out]	INFO	INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = i, and i is: <= N the Cholesky factorization of the matrix could not be performed because the leading principal minor of order i was not positive. > N the SVD algorithm failed to converge; if INFO = N+i, i off-diagonal elements of the bidiagonal factor did not converge to zero.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 144 of file spteqr.f.

*
*  -- LAPACK computational 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          COMPZ
      INTEGER            INFO, LDZ, N
*     ..
*     .. Array Arguments ..
      REAL               D( * ), E( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      parameter( zero = 0.0e0, one = 1.0e0 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           lsame
*     ..
*     .. External Subroutines ..
      EXTERNAL           sbdsqr, slaset, spttrf, xerbla
*     ..
*     .. Local Arrays ..
      REAL               C( 1, 1 ), VT( 1, 1 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, ICOMPZ, NRU
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, sqrt
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
*
      IF( lsame( compz, 'N' ) ) THEN
         icompz = 0
      ELSE IF( lsame( compz, 'V' ) ) THEN
         icompz = 1
      ELSE IF( lsame( compz, 'I' ) ) THEN
         icompz = 2
      ELSE
         icompz = -1
      END IF
      IF( icompz.LT.0 ) THEN
         info = -1
      ELSE IF( n.LT.0 ) THEN
         info = -2
      ELSE IF( ( ldz.LT.1 ) .OR. ( icompz.GT.0 .AND. ldz.LT.max( 1,
     $         n ) ) ) THEN
         info = -6
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SPTEQR', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 )
     $   RETURN
*
      IF( n.EQ.1 ) THEN
         IF( icompz.GT.0 )
     $      z( 1, 1 ) = one
         RETURN
      END IF
      IF( icompz.EQ.2 )
     $   CALL slaset( 'Full', n, n, zero, one, z, ldz )
*
*     Call SPTTRF to factor the matrix.
*
      CALL spttrf( n, d, e, info )
      IF( info.NE.0 )
     $   RETURN
      DO 10 i = 1, n
         d( i ) = sqrt( d( i ) )
   10 CONTINUE
      DO 20 i = 1, n - 1
         e( i ) = e( i )*d( i )
   20 CONTINUE
*
*     Call SBDSQR to compute the singular values/vectors of the
*     bidiagonal factor.
*
      IF( icompz.GT.0 ) THEN
         nru = n
      ELSE
         nru = 0
      END IF
      CALL sbdsqr( 'Lower', n, 0, nru, 0, d, e, vt, 1, z, ldz, c, 1,
     $             work, info )
*
*     Square the singular values.
*
      IF( info.EQ.0 ) THEN
         DO 30 i = 1, n
            d( i ) = d( i )*d( i )
   30    CONTINUE
      ELSE
         info = n + info
      END IF
*
      RETURN
*
*     End of SPTEQR
*

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