LAPACK  3.9.0 LAPACK: Linear Algebra PACKage
zpteqr.f
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1 *> \brief \b ZPTEQR
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
22 *
23 * .. Scalar Arguments ..
24 * CHARACTER COMPZ
25 * INTEGER INFO, LDZ, N
26 * ..
27 * .. Array Arguments ..
28 * DOUBLE PRECISION D( * ), E( * ), WORK( * )
29 * COMPLEX*16 Z( LDZ, * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> ZPTEQR computes all eigenvalues and, optionally, eigenvectors of a
39 *> symmetric positive definite tridiagonal matrix by first factoring the
40 *> matrix using DPTTRF and then calling ZBDSQR to compute the singular
41 *> values of the bidiagonal factor.
42 *>
43 *> This routine computes the eigenvalues of the positive definite
44 *> tridiagonal matrix to high relative accuracy. This means that if the
45 *> eigenvalues range over many orders of magnitude in size, then the
46 *> small eigenvalues and corresponding eigenvectors will be computed
47 *> more accurately than, for example, with the standard QR method.
48 *>
49 *> The eigenvectors of a full or band positive definite Hermitian matrix
50 *> can also be found if ZHETRD, ZHPTRD, or ZHBTRD has been used to
51 *> reduce this matrix to tridiagonal form. (The reduction to
52 *> tridiagonal form, however, may preclude the possibility of obtaining
53 *> high relative accuracy in the small eigenvalues of the original
54 *> matrix, if these eigenvalues range over many orders of magnitude.)
55 *> \endverbatim
56 *
57 * Arguments:
58 * ==========
59 *
60 *> \param[in] COMPZ
61 *> \verbatim
62 *> COMPZ is CHARACTER*1
63 *> = 'N': Compute eigenvalues only.
64 *> = 'V': Compute eigenvectors of original Hermitian
65 *> matrix also. Array Z contains the unitary matrix
66 *> used to reduce the original matrix to tridiagonal
67 *> form.
68 *> = 'I': Compute eigenvectors of tridiagonal matrix also.
69 *> \endverbatim
70 *>
71 *> \param[in] N
72 *> \verbatim
73 *> N is INTEGER
74 *> The order of the matrix. N >= 0.
75 *> \endverbatim
76 *>
77 *> \param[in,out] D
78 *> \verbatim
79 *> D is DOUBLE PRECISION array, dimension (N)
80 *> On entry, the n diagonal elements of the tridiagonal matrix.
81 *> On normal exit, D contains the eigenvalues, in descending
82 *> order.
83 *> \endverbatim
84 *>
85 *> \param[in,out] E
86 *> \verbatim
87 *> E is DOUBLE PRECISION array, dimension (N-1)
88 *> On entry, the (n-1) subdiagonal elements of the tridiagonal
89 *> matrix.
90 *> On exit, E has been destroyed.
91 *> \endverbatim
92 *>
93 *> \param[in,out] Z
94 *> \verbatim
95 *> Z is COMPLEX*16 array, dimension (LDZ, N)
96 *> On entry, if COMPZ = 'V', the unitary matrix used in the
97 *> reduction to tridiagonal form.
98 *> On exit, if COMPZ = 'V', the orthonormal eigenvectors of the
99 *> original Hermitian matrix;
100 *> if COMPZ = 'I', the orthonormal eigenvectors of the
101 *> tridiagonal matrix.
102 *> If INFO > 0 on exit, Z contains the eigenvectors associated
103 *> with only the stored eigenvalues.
104 *> If COMPZ = 'N', then Z is not referenced.
105 *> \endverbatim
106 *>
107 *> \param[in] LDZ
108 *> \verbatim
109 *> LDZ is INTEGER
110 *> The leading dimension of the array Z. LDZ >= 1, and if
111 *> COMPZ = 'V' or 'I', LDZ >= max(1,N).
112 *> \endverbatim
113 *>
114 *> \param[out] WORK
115 *> \verbatim
116 *> WORK is DOUBLE PRECISION array, dimension (4*N)
117 *> \endverbatim
118 *>
119 *> \param[out] INFO
120 *> \verbatim
121 *> INFO is INTEGER
122 *> = 0: successful exit.
123 *> < 0: if INFO = -i, the i-th argument had an illegal value.
124 *> > 0: if INFO = i, and i is:
125 *> <= N the Cholesky factorization of the matrix could
126 *> not be performed because the i-th principal minor
127 *> was not positive definite.
128 *> > N the SVD algorithm failed to converge;
129 *> if INFO = N+i, i off-diagonal elements of the
130 *> bidiagonal factor did not converge to zero.
131 *> \endverbatim
132 *
133 * Authors:
134 * ========
135 *
136 *> \author Univ. of Tennessee
137 *> \author Univ. of California Berkeley
138 *> \author Univ. of Colorado Denver
139 *> \author NAG Ltd.
140 *
141 *> \date December 2016
142 *
143 *> \ingroup complex16PTcomputational
144 *
145 * =====================================================================
146  SUBROUTINE zpteqr( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
147 *
148 * -- LAPACK computational routine (version 3.7.0) --
149 * -- LAPACK is a software package provided by Univ. of Tennessee, --
150 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
151 * December 2016
152 *
153 * .. Scalar Arguments ..
154  CHARACTER COMPZ
155  INTEGER INFO, LDZ, N
156 * ..
157 * .. Array Arguments ..
158  DOUBLE PRECISION D( * ), E( * ), WORK( * )
159  COMPLEX*16 Z( LDZ, * )
160 * ..
161 *
162 * ====================================================================
163 *
164 * .. Parameters ..
165  COMPLEX*16 CZERO, CONE
166  parameter( czero = ( 0.0d+0, 0.0d+0 ),
167  \$ cone = ( 1.0d+0, 0.0d+0 ) )
168 * ..
169 * .. External Functions ..
170  LOGICAL LSAME
171  EXTERNAL lsame
172 * ..
173 * .. External Subroutines ..
174  EXTERNAL dpttrf, xerbla, zbdsqr, zlaset
175 * ..
176 * .. Local Arrays ..
177  COMPLEX*16 C( 1, 1 ), VT( 1, 1 )
178 * ..
179 * .. Local Scalars ..
180  INTEGER I, ICOMPZ, NRU
181 * ..
182 * .. Intrinsic Functions ..
183  INTRINSIC max, sqrt
184 * ..
185 * .. Executable Statements ..
186 *
187 * Test the input parameters.
188 *
189  info = 0
190 *
191  IF( lsame( compz, 'N' ) ) THEN
192  icompz = 0
193  ELSE IF( lsame( compz, 'V' ) ) THEN
194  icompz = 1
195  ELSE IF( lsame( compz, 'I' ) ) THEN
196  icompz = 2
197  ELSE
198  icompz = -1
199  END IF
200  IF( icompz.LT.0 ) THEN
201  info = -1
202  ELSE IF( n.LT.0 ) THEN
203  info = -2
204  ELSE IF( ( ldz.LT.1 ) .OR. ( icompz.GT.0 .AND. ldz.LT.max( 1,
205  \$ n ) ) ) THEN
206  info = -6
207  END IF
208  IF( info.NE.0 ) THEN
209  CALL xerbla( 'ZPTEQR', -info )
210  RETURN
211  END IF
212 *
213 * Quick return if possible
214 *
215  IF( n.EQ.0 )
216  \$ RETURN
217 *
218  IF( n.EQ.1 ) THEN
219  IF( icompz.GT.0 )
220  \$ z( 1, 1 ) = cone
221  RETURN
222  END IF
223  IF( icompz.EQ.2 )
224  \$ CALL zlaset( 'Full', n, n, czero, cone, z, ldz )
225 *
226 * Call DPTTRF to factor the matrix.
227 *
228  CALL dpttrf( n, d, e, info )
229  IF( info.NE.0 )
230  \$ RETURN
231  DO 10 i = 1, n
232  d( i ) = sqrt( d( i ) )
233  10 CONTINUE
234  DO 20 i = 1, n - 1
235  e( i ) = e( i )*d( i )
236  20 CONTINUE
237 *
238 * Call ZBDSQR to compute the singular values/vectors of the
239 * bidiagonal factor.
240 *
241  IF( icompz.GT.0 ) THEN
242  nru = n
243  ELSE
244  nru = 0
245  END IF
246  CALL zbdsqr( 'Lower', n, 0, nru, 0, d, e, vt, 1, z, ldz, c, 1,
247  \$ work, info )
248 *
249 * Square the singular values.
250 *
251  IF( info.EQ.0 ) THEN
252  DO 30 i = 1, n
253  d( i ) = d( i )*d( i )
254  30 CONTINUE
255  ELSE
256  info = n + info
257  END IF
258 *
259  RETURN
260 *
261 * End of ZPTEQR
262 *
263  END
zbdsqr
subroutine zbdsqr(UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, RWORK, INFO)
ZBDSQR
Definition: zbdsqr.f:224
dpttrf
subroutine dpttrf(N, D, E, INFO)
DPTTRF
Definition: dpttrf.f:93
zpteqr
subroutine zpteqr(COMPZ, N, D, E, Z, LDZ, WORK, INFO)
ZPTEQR
Definition: zpteqr.f:147
zlaset
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:108
xerbla
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62