LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ 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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