1: *> \brief \b DLAED1 used by DSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLAED1 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed1.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed1.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed1.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK,
22: * INFO )
23: *
24: * .. Scalar Arguments ..
25: * INTEGER CUTPNT, INFO, LDQ, N
26: * DOUBLE PRECISION RHO
27: * ..
28: * .. Array Arguments ..
29: * INTEGER INDXQ( * ), IWORK( * )
30: * DOUBLE PRECISION D( * ), Q( LDQ, * ), WORK( * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *>
39: *> DLAED1 computes the updated eigensystem of a diagonal
40: *> matrix after modification by a rank-one symmetric matrix. This
41: *> routine is used only for the eigenproblem which requires all
42: *> eigenvalues and eigenvectors of a tridiagonal matrix. DLAED7 handles
43: *> the case in which eigenvalues only or eigenvalues and eigenvectors
44: *> of a full symmetric matrix (which was reduced to tridiagonal form)
45: *> are desired.
46: *>
47: *> T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
48: *>
49: *> where Z = Q**T*u, u is a vector of length N with ones in the
50: *> CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
51: *>
52: *> The eigenvectors of the original matrix are stored in Q, and the
53: *> eigenvalues are in D. The algorithm consists of three stages:
54: *>
55: *> The first stage consists of deflating the size of the problem
56: *> when there are multiple eigenvalues or if there is a zero in
57: *> the Z vector. For each such occurrence the dimension of the
58: *> secular equation problem is reduced by one. This stage is
59: *> performed by the routine DLAED2.
60: *>
61: *> The second stage consists of calculating the updated
62: *> eigenvalues. This is done by finding the roots of the secular
63: *> equation via the routine DLAED4 (as called by DLAED3).
64: *> This routine also calculates the eigenvectors of the current
65: *> problem.
66: *>
67: *> The final stage consists of computing the updated eigenvectors
68: *> directly using the updated eigenvalues. The eigenvectors for
69: *> the current problem are multiplied with the eigenvectors from
70: *> the overall problem.
71: *> \endverbatim
72: *
73: * Arguments:
74: * ==========
75: *
76: *> \param[in] N
77: *> \verbatim
78: *> N is INTEGER
79: *> The dimension of the symmetric tridiagonal matrix. N >= 0.
80: *> \endverbatim
81: *>
82: *> \param[in,out] D
83: *> \verbatim
84: *> D is DOUBLE PRECISION array, dimension (N)
85: *> On entry, the eigenvalues of the rank-1-perturbed matrix.
86: *> On exit, the eigenvalues of the repaired matrix.
87: *> \endverbatim
88: *>
89: *> \param[in,out] Q
90: *> \verbatim
91: *> Q is DOUBLE PRECISION array, dimension (LDQ,N)
92: *> On entry, the eigenvectors of the rank-1-perturbed matrix.
93: *> On exit, the eigenvectors of the repaired tridiagonal matrix.
94: *> \endverbatim
95: *>
96: *> \param[in] LDQ
97: *> \verbatim
98: *> LDQ is INTEGER
99: *> The leading dimension of the array Q. LDQ >= max(1,N).
100: *> \endverbatim
101: *>
102: *> \param[in,out] INDXQ
103: *> \verbatim
104: *> INDXQ is INTEGER array, dimension (N)
105: *> On entry, the permutation which separately sorts the two
106: *> subproblems in D into ascending order.
107: *> On exit, the permutation which will reintegrate the
108: *> subproblems back into sorted order,
109: *> i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.
110: *> \endverbatim
111: *>
112: *> \param[in] RHO
113: *> \verbatim
114: *> RHO is DOUBLE PRECISION
115: *> The subdiagonal entry used to create the rank-1 modification.
116: *> \endverbatim
117: *>
118: *> \param[in] CUTPNT
119: *> \verbatim
120: *> CUTPNT is INTEGER
121: *> The location of the last eigenvalue in the leading sub-matrix.
122: *> min(1,N) <= CUTPNT <= N/2.
123: *> \endverbatim
124: *>
125: *> \param[out] WORK
126: *> \verbatim
127: *> WORK is DOUBLE PRECISION array, dimension (4*N + N**2)
128: *> \endverbatim
129: *>
130: *> \param[out] IWORK
131: *> \verbatim
132: *> IWORK is INTEGER array, dimension (4*N)
133: *> \endverbatim
134: *>
135: *> \param[out] INFO
136: *> \verbatim
137: *> INFO is INTEGER
138: *> = 0: successful exit.
139: *> < 0: if INFO = -i, the i-th argument had an illegal value.
140: *> > 0: if INFO = 1, an eigenvalue did not converge
141: *> \endverbatim
142: *
143: * Authors:
144: * ========
145: *
146: *> \author Univ. of Tennessee
147: *> \author Univ. of California Berkeley
148: *> \author Univ. of Colorado Denver
149: *> \author NAG Ltd.
150: *
151: *> \ingroup auxOTHERcomputational
152: *
153: *> \par Contributors:
154: * ==================
155: *>
156: *> Jeff Rutter, Computer Science Division, University of California
157: *> at Berkeley, USA \n
158: *> Modified by Francoise Tisseur, University of Tennessee
159: *>
160: * =====================================================================
161: SUBROUTINE DLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK,
162: $ INFO )
163: *
164: * -- LAPACK computational routine --
165: * -- LAPACK is a software package provided by Univ. of Tennessee, --
166: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
167: *
168: * .. Scalar Arguments ..
169: INTEGER CUTPNT, INFO, LDQ, N
170: DOUBLE PRECISION RHO
171: * ..
172: * .. Array Arguments ..
173: INTEGER INDXQ( * ), IWORK( * )
174: DOUBLE PRECISION D( * ), Q( LDQ, * ), WORK( * )
175: * ..
176: *
177: * =====================================================================
178: *
179: * .. Local Scalars ..
180: INTEGER COLTYP, I, IDLMDA, INDX, INDXC, INDXP, IQ2, IS,
181: $ IW, IZ, K, N1, N2, ZPP1
182: * ..
183: * .. External Subroutines ..
184: EXTERNAL DCOPY, DLAED2, DLAED3, DLAMRG, XERBLA
185: * ..
186: * .. Intrinsic Functions ..
187: INTRINSIC MAX, MIN
188: * ..
189: * .. Executable Statements ..
190: *
191: * Test the input parameters.
192: *
193: INFO = 0
194: *
195: IF( N.LT.0 ) THEN
196: INFO = -1
197: ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
198: INFO = -4
199: ELSE IF( MIN( 1, N / 2 ).GT.CUTPNT .OR. ( N / 2 ).LT.CUTPNT ) THEN
200: INFO = -7
201: END IF
202: IF( INFO.NE.0 ) THEN
203: CALL XERBLA( 'DLAED1', -INFO )
204: RETURN
205: END IF
206: *
207: * Quick return if possible
208: *
209: IF( N.EQ.0 )
210: $ RETURN
211: *
212: * The following values are integer pointers which indicate
213: * the portion of the workspace
214: * used by a particular array in DLAED2 and DLAED3.
215: *
216: IZ = 1
217: IDLMDA = IZ + N
218: IW = IDLMDA + N
219: IQ2 = IW + N
220: *
221: INDX = 1
222: INDXC = INDX + N
223: COLTYP = INDXC + N
224: INDXP = COLTYP + N
225: *
226: *
227: * Form the z-vector which consists of the last row of Q_1 and the
228: * first row of Q_2.
229: *
230: CALL DCOPY( CUTPNT, Q( CUTPNT, 1 ), LDQ, WORK( IZ ), 1 )
231: ZPP1 = CUTPNT + 1
232: CALL DCOPY( N-CUTPNT, Q( ZPP1, ZPP1 ), LDQ, WORK( IZ+CUTPNT ), 1 )
233: *
234: * Deflate eigenvalues.
235: *
236: CALL DLAED2( K, N, CUTPNT, D, Q, LDQ, INDXQ, RHO, WORK( IZ ),
237: $ WORK( IDLMDA ), WORK( IW ), WORK( IQ2 ),
238: $ IWORK( INDX ), IWORK( INDXC ), IWORK( INDXP ),
239: $ IWORK( COLTYP ), INFO )
240: *
241: IF( INFO.NE.0 )
242: $ GO TO 20
243: *
244: * Solve Secular Equation.
245: *
246: IF( K.NE.0 ) THEN
247: IS = ( IWORK( COLTYP )+IWORK( COLTYP+1 ) )*CUTPNT +
248: $ ( IWORK( COLTYP+1 )+IWORK( COLTYP+2 ) )*( N-CUTPNT ) + IQ2
249: CALL DLAED3( K, N, CUTPNT, D, Q, LDQ, RHO, WORK( IDLMDA ),
250: $ WORK( IQ2 ), IWORK( INDXC ), IWORK( COLTYP ),
251: $ WORK( IW ), WORK( IS ), INFO )
252: IF( INFO.NE.0 )
253: $ GO TO 20
254: *
255: * Prepare the INDXQ sorting permutation.
256: *
257: N1 = K
258: N2 = N - K
259: CALL DLAMRG( N1, N2, D, 1, -1, INDXQ )
260: ELSE
261: DO 10 I = 1, N
262: INDXQ( I ) = I
263: 10 CONTINUE
264: END IF
265: *
266: 20 CONTINUE
267: RETURN
268: *
269: * End of DLAED1
270: *
271: END
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