1: *> \brief \b DPTEQR
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DPTEQR + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpteqr.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpteqr.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpteqr.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DPTEQR( 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( * ), Z( LDZ, * )
29: * ..
30: *
31: *
32: *> \par Purpose:
33: * =============
34: *>
35: *> \verbatim
36: *>
37: *> DPTEQR computes all eigenvalues and, optionally, eigenvectors of a
38: *> symmetric positive definite tridiagonal matrix by first factoring the
39: *> matrix using DPTTRF, and then calling DBDSQR to compute the singular
40: *> values of the bidiagonal factor.
41: *>
42: *> This routine computes the eigenvalues of the positive definite
43: *> tridiagonal matrix to high relative accuracy. This means that if the
44: *> eigenvalues range over many orders of magnitude in size, then the
45: *> small eigenvalues and corresponding eigenvectors will be computed
46: *> more accurately than, for example, with the standard QR method.
47: *>
48: *> The eigenvectors of a full or band symmetric positive definite matrix
49: *> can also be found if DSYTRD, DSPTRD, or DSBTRD has been used to
50: *> reduce this matrix to tridiagonal form. (The reduction to tridiagonal
51: *> form, however, may preclude the possibility of obtaining high
52: *> relative accuracy in the small eigenvalues of the original matrix, if
53: *> these eigenvalues range over many orders of magnitude.)
54: *> \endverbatim
55: *
56: * Arguments:
57: * ==========
58: *
59: *> \param[in] COMPZ
60: *> \verbatim
61: *> COMPZ is CHARACTER*1
62: *> = 'N': Compute eigenvalues only.
63: *> = 'V': Compute eigenvectors of original symmetric
64: *> matrix also. Array Z contains the orthogonal
65: *> matrix used to reduce the original matrix to
66: *> tridiagonal form.
67: *> = 'I': Compute eigenvectors of tridiagonal matrix also.
68: *> \endverbatim
69: *>
70: *> \param[in] N
71: *> \verbatim
72: *> N is INTEGER
73: *> The order of the matrix. N >= 0.
74: *> \endverbatim
75: *>
76: *> \param[in,out] D
77: *> \verbatim
78: *> D is DOUBLE PRECISION array, dimension (N)
79: *> On entry, the n diagonal elements of the tridiagonal
80: *> 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 DOUBLE PRECISION array, dimension (LDZ, N)
96: *> On entry, if COMPZ = 'V', the orthogonal matrix used in the
97: *> reduction to tridiagonal form.
98: *> On exit, if COMPZ = 'V', the orthonormal eigenvectors of the
99: *> original symmetric 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: *> \ingroup doublePTcomputational
142: *
143: * =====================================================================
144: SUBROUTINE DPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
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( * ), Z( LDZ, * )
156: * ..
157: *
158: * =====================================================================
159: *
160: * .. Parameters ..
161: DOUBLE PRECISION ZERO, ONE
162: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
163: * ..
164: * .. External Functions ..
165: LOGICAL LSAME
166: EXTERNAL LSAME
167: * ..
168: * .. External Subroutines ..
169: EXTERNAL DBDSQR, DLASET, DPTTRF, XERBLA
170: * ..
171: * .. Local Arrays ..
172: DOUBLE PRECISION 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( 'DPTEQR', -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 DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
220: *
221: * Call DPTTRF to factor the matrix.
222: *
223: CALL DPTTRF( 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 DBDSQR 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 DBDSQR( '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 DPTEQR
257: *
258: END
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