 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ 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

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 i-th principal minor was not positive definite. > N the SVD algorithm failed to converge; if INFO = N+i, i off-diagonal elements of the bidiagonal factor did not converge to zero.```

Definition at line 144 of file spteqr.f.

145 *
146 * -- LAPACK computational routine --
147 * -- LAPACK is a software package provided by Univ. of Tennessee, --
148 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
149 *
150 * .. Scalar Arguments ..
151  CHARACTER COMPZ
152  INTEGER INFO, LDZ, N
153 * ..
154 * .. Array Arguments ..
155  REAL D( * ), E( * ), WORK( * ), Z( LDZ, * )
156 * ..
157 *
158 * =====================================================================
159 *
160 * .. Parameters ..
161  REAL ZERO, ONE
162  parameter( zero = 0.0e0, one = 1.0e0 )
163 * ..
164 * .. External Functions ..
165  LOGICAL LSAME
166  EXTERNAL lsame
167 * ..
168 * .. External Subroutines ..
169  EXTERNAL sbdsqr, slaset, spttrf, xerbla
170 * ..
171 * .. Local Arrays ..
172  REAL C( 1, 1 ), VT( 1, 1 )
173 * ..
174 * .. Local Scalars ..
175  INTEGER I, ICOMPZ, NRU
176 * ..
177 * .. Intrinsic Functions ..
178  INTRINSIC max, sqrt
179 * ..
180 * .. Executable Statements ..
181 *
182 * Test the input parameters.
183 *
184  info = 0
185 *
186  IF( lsame( compz, 'N' ) ) THEN
187  icompz = 0
188  ELSE IF( lsame( compz, 'V' ) ) THEN
189  icompz = 1
190  ELSE IF( lsame( compz, 'I' ) ) THEN
191  icompz = 2
192  ELSE
193  icompz = -1
194  END IF
195  IF( icompz.LT.0 ) THEN
196  info = -1
197  ELSE IF( n.LT.0 ) THEN
198  info = -2
199  ELSE IF( ( ldz.LT.1 ) .OR. ( icompz.GT.0 .AND. ldz.LT.max( 1,
200  \$ n ) ) ) THEN
201  info = -6
202  END IF
203  IF( info.NE.0 ) THEN
204  CALL xerbla( 'SPTEQR', -info )
205  RETURN
206  END IF
207 *
208 * Quick return if possible
209 *
210  IF( n.EQ.0 )
211  \$ RETURN
212 *
213  IF( n.EQ.1 ) THEN
214  IF( icompz.GT.0 )
215  \$ z( 1, 1 ) = one
216  RETURN
217  END IF
218  IF( icompz.EQ.2 )
219  \$ CALL slaset( 'Full', n, n, zero, one, z, ldz )
220 *
221 * Call SPTTRF to factor the matrix.
222 *
223  CALL spttrf( n, d, e, info )
224  IF( info.NE.0 )
225  \$ RETURN
226  DO 10 i = 1, n
227  d( i ) = sqrt( d( i ) )
228  10 CONTINUE
229  DO 20 i = 1, n - 1
230  e( i ) = e( i )*d( i )
231  20 CONTINUE
232 *
233 * Call SBDSQR to compute the singular values/vectors of the
234 * bidiagonal factor.
235 *
236  IF( icompz.GT.0 ) THEN
237  nru = n
238  ELSE
239  nru = 0
240  END IF
241  CALL sbdsqr( 'Lower', n, 0, nru, 0, d, e, vt, 1, z, ldz, c, 1,
242  \$ work, info )
243 *
244 * Square the singular values.
245 *
246  IF( info.EQ.0 ) THEN
247  DO 30 i = 1, n
248  d( i ) = d( i )*d( i )
249  30 CONTINUE
250  ELSE
251  info = n + info
252  END IF
253 *
254  RETURN
255 *
256 * End of SPTEQR
257 *
subroutine slaset(UPLO, M, N, ALPHA, BETA, A, LDA)
SLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: slaset.f:110
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine spttrf(N, D, E, INFO)
SPTTRF
Definition: spttrf.f:91
subroutine sbdsqr(UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, WORK, INFO)
SBDSQR
Definition: sbdsqr.f:240
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