LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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◆ 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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