1ZLAED7(1) LAPACK routine (version 3.2) ZLAED7(1)
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6 ZLAED7 - computes the updated eigensystem of a diagonal matrix after
7 modification by a rank-one symmetric matrix
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10 SUBROUTINE ZLAED7( N, CUTPNT, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, LDQ,
11 RHO, INDXQ, QSTORE, QPTR, PRMPTR, PERM, GIVPTR, GIV‐
12 COL, GIVNUM, WORK, RWORK, IWORK, INFO )
13 INTEGER CURLVL, CURPBM, CUTPNT, INFO, LDQ, N, QSIZ, TLVLS
15 DOUBLE PRECISION RHO
17 INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ), IWORK( * ),
19 PERM( * ), PRMPTR( * ), QPTR( * )
20 DOUBLE PRECISION D( * ), GIVNUM( 2, * ), QSTORE( * ),
22 RWORK( * )
23 COMPLEX*16 Q( LDQ, * ), WORK( * )
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27 ZLAED7 computes the updated eigensystem of a diagonal matrix after mod‐
28 ification by a rank-one symmetric matrix. This routine is used only for
29 the eigenproblem which requires all eigenvalues and optionally eigen‐
30 vectors of a dense or banded Hermitian matrix that has been reduced to
31 tridiagonal form.
32 T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out)
33 where Z = Q'u, u is a vector of length N with ones in the
34 CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
35 The eigenvectors of the original matrix are stored in Q, and the
36 eigenvalues are in D. The algorithm consists of three stages:
37 The first stage consists of deflating the size of the problem
38 when there are multiple eigenvalues or if there is a zero in
39 the Z vector. For each such occurence the dimension of the
40 secular equation problem is reduced by one. This stage is
41 performed by the routine DLAED2.
42 The second stage consists of calculating the updated
43 eigenvalues. This is done by finding the roots of the secular
44 equation via the routine DLAED4 (as called by SLAED3).
45 This routine also calculates the eigenvectors of the current
46 problem.
47 The final stage consists of computing the updated eigenvectors
48 directly using the updated eigenvalues. The eigenvectors for
49 the current problem are multiplied with the eigenvectors from
50 the overall problem.
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53 N (input) INTEGER
54 The dimension of the symmetric tridiagonal matrix. N >= 0.
55 CUTPNT (input) INTEGER Contains the location of the last eigen‐
56 value in the leading sub-matrix. min(1,N) <= CUTPNT <= N.
57 QSIZ (input) INTEGER
59 The dimension of the unitary matrix used to reduce the full
60 matrix to tridiagonal form. QSIZ >= N.
61 TLVLS (input) INTEGER
63 The total number of merging levels in the overall divide and
64 conquer tree. CURLVL (input) INTEGER The current level in the
65 overall merge routine, 0 <= curlvl <= tlvls. CURPBM (input)
66 INTEGER The current problem in the current level in the overall
67 merge routine (counting from upper left to lower right).
68 D (input/output) DOUBLE PRECISION array, dimension (N)
70 On entry, the eigenvalues of the rank-1-perturbed matrix. On
71 exit, the eigenvalues of the repaired matrix.
72 Q (input/output) COMPLEX*16 array, dimension (LDQ,N)
74 On entry, the eigenvectors of the rank-1-perturbed matrix. On
75 exit, the eigenvectors of the repaired tridiagonal matrix.
76 LDQ (input) INTEGER
78 The leading dimension of the array Q. LDQ >= max(1,N).
79 RHO (input) DOUBLE PRECISION
81 Contains the subdiagonal element used to create the rank-1 modi‐
82 fication.
83 INDXQ (output) INTEGER array, dimension (N)
85 This contains the permutation which will reintegrate the sub‐
86 problem just solved back into sorted order, ie. D( INDXQ( I = 1,
87 N ) ) will be in ascending order.
88 IWORK (workspace) INTEGER array, dimension (4*N)
90 RWORK (workspace) DOUBLE PRECISION array,
92 dimension (3*N+2*QSIZ*N)
93 WORK (workspace) COMPLEX*16 array, dimension (QSIZ*N)
95 QSTORE (input/output) DOUBLE PRECISION array, dimension (N**2+1)
96 Stores eigenvectors of submatrices encountered during divide and
97 conquer, packed together. QPTR points to beginning of the subma‐
98 trices.
99 QPTR (input/output) INTEGER array, dimension (N+2)
101 List of indices pointing to beginning of submatrices stored in
102 QSTORE. The submatrices are numbered starting at the bottom left
103 of the divide and conquer tree, from left to right and bottom to
104 top. PRMPTR (input) INTEGER array, dimension (N lg N) Contains
105 a list of pointers which indicate where in PERM a level's permu‐
106 tation is stored. PRMPTR(i+1) - PRMPTR(i) indicates the size of
107 the permutation and also the size of the full, non-deflated
108 problem.
109 PERM (input) INTEGER array, dimension (N lg N)
111 Contains the permutations (from deflation and sorting) to be
112 applied to each eigenblock. GIVPTR (input) INTEGER array,
113 dimension (N lg N) Contains a list of pointers which indicate
114 where in GIVCOL a level's Givens rotations are stored.
115 GIVPTR(i+1) - GIVPTR(i) indicates the number of Givens rota‐
116 tions. GIVCOL (input) INTEGER array, dimension (2, N lg N) Each
117 pair of numbers indicates a pair of columns to take place in a
118 Givens rotation. GIVNUM (input) DOUBLE PRECISION array, dimen‐
119 sion (2, N lg N) Each number indicates the S value to be used in
120 the corresponding Givens rotation.
121 INFO (output) INTEGER
123 = 0: successful exit.
124 < 0: if INFO = -i, the i-th argument had an illegal value.
125 > 0: if INFO = 1, an eigenvalue did not converge
126 LAPACK routine (version 3.2) November 2008 ZLAED7(1)