 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ zpteqr()

 subroutine zpteqr ( character COMPZ, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer INFO )

ZPTEQR

Purpose:
``` ZPTEQR computes all eigenvalues and, optionally, eigenvectors of a
symmetric positive definite tridiagonal matrix by first factoring the
matrix using DPTTRF and then calling ZBDSQR 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 positive definite Hermitian matrix
can also be found if ZHETRD, ZHPTRD, or ZHBTRD 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 Hermitian matrix also. Array Z contains the unitary 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 COMPLEX*16 array, dimension (LDZ, N) On entry, if COMPZ = 'V', the unitary matrix used in the reduction to tridiagonal form. On exit, if COMPZ = 'V', the orthonormal eigenvectors of the original Hermitian 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 DOUBLE PRECISION 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 zpteqr.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  DOUBLE PRECISION D( * ), E( * ), WORK( * )
156  COMPLEX*16 Z( LDZ, * )
157 * ..
158 *
159 * ====================================================================
160 *
161 * .. Parameters ..
162  COMPLEX*16 CZERO, CONE
163  parameter( czero = ( 0.0d+0, 0.0d+0 ),
164  \$ cone = ( 1.0d+0, 0.0d+0 ) )
165 * ..
166 * .. External Functions ..
167  LOGICAL LSAME
168  EXTERNAL lsame
169 * ..
170 * .. External Subroutines ..
171  EXTERNAL dpttrf, xerbla, zbdsqr, zlaset
172 * ..
173 * .. Local Arrays ..
174  COMPLEX*16 C( 1, 1 ), VT( 1, 1 )
175 * ..
176 * .. Local Scalars ..
177  INTEGER I, ICOMPZ, NRU
178 * ..
179 * .. Intrinsic Functions ..
180  INTRINSIC max, sqrt
181 * ..
182 * .. Executable Statements ..
183 *
184 * Test the input parameters.
185 *
186  info = 0
187 *
188  IF( lsame( compz, 'N' ) ) THEN
189  icompz = 0
190  ELSE IF( lsame( compz, 'V' ) ) THEN
191  icompz = 1
192  ELSE IF( lsame( compz, 'I' ) ) THEN
193  icompz = 2
194  ELSE
195  icompz = -1
196  END IF
197  IF( icompz.LT.0 ) THEN
198  info = -1
199  ELSE IF( n.LT.0 ) THEN
200  info = -2
201  ELSE IF( ( ldz.LT.1 ) .OR. ( icompz.GT.0 .AND. ldz.LT.max( 1,
202  \$ n ) ) ) THEN
203  info = -6
204  END IF
205  IF( info.NE.0 ) THEN
206  CALL xerbla( 'ZPTEQR', -info )
207  RETURN
208  END IF
209 *
210 * Quick return if possible
211 *
212  IF( n.EQ.0 )
213  \$ RETURN
214 *
215  IF( n.EQ.1 ) THEN
216  IF( icompz.GT.0 )
217  \$ z( 1, 1 ) = cone
218  RETURN
219  END IF
220  IF( icompz.EQ.2 )
221  \$ CALL zlaset( 'Full', n, n, czero, cone, z, ldz )
222 *
223 * Call DPTTRF to factor the matrix.
224 *
225  CALL dpttrf( n, d, e, info )
226  IF( info.NE.0 )
227  \$ RETURN
228  DO 10 i = 1, n
229  d( i ) = sqrt( d( i ) )
230  10 CONTINUE
231  DO 20 i = 1, n - 1
232  e( i ) = e( i )*d( i )
233  20 CONTINUE
234 *
235 * Call ZBDSQR to compute the singular values/vectors of the
236 * bidiagonal factor.
237 *
238  IF( icompz.GT.0 ) THEN
239  nru = n
240  ELSE
241  nru = 0
242  END IF
243  CALL zbdsqr( 'Lower', n, 0, nru, 0, d, e, vt, 1, z, ldz, c, 1,
244  \$ work, info )
245 *
246 * Square the singular values.
247 *
248  IF( info.EQ.0 ) THEN
249  DO 30 i = 1, n
250  d( i ) = d( i )*d( i )
251  30 CONTINUE
252  ELSE
253  info = n + info
254  END IF
255 *
256  RETURN
257 *
258 * End of ZPTEQR
259 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine zlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
ZLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: zlaset.f:106
subroutine zbdsqr(UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, RWORK, INFO)
ZBDSQR
Definition: zbdsqr.f:222
subroutine dpttrf(N, D, E, INFO)
DPTTRF
Definition: dpttrf.f:91
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