1SLAED7(1) LAPACK routine (version 3.2) SLAED7(1)
2
6 SLAED7 - computes the updated eigensystem of a diagonal matrix after
7 modification by a rank-one symmetric matrix
8
10 SUBROUTINE SLAED7( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, LDQ,
11 INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR, PERM,
12 GIVPTR, GIVCOL, GIVNUM, WORK, IWORK, INFO )
13 INTEGER CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N, QSIZ,
15 TLVLS
16 REAL RHO
18 INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ), IWORK( * ),
20 PERM( * ), PRMPTR( * ), QPTR( * )
21 REAL D( * ), GIVNUM( 2, * ), Q( LDQ, * ), QSTORE( * ),
23 WORK( * )
24
26 SLAED7 computes the updated eigensystem of a diagonal matrix after mod‐
27 ification by a rank-one symmetric matrix. This routine is used only for
28 the eigenproblem which requires all eigenvalues and optionally eigen‐
29 vectors of a dense symmetric matrix that has been reduced to tridiago‐
30 nal form. SLAED1 handles the case in which all eigenvalues and eigen‐
31 vectors of a symmetric tridiagonal matrix are desired.
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 SLAED8.
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 SLAED4 (as called by SLAED9).
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.
51
53 ICOMPQ (input) INTEGER
54 = 0: Compute eigenvalues only.
55 = 1: Compute eigenvectors of original dense symmetric matrix
56 also. On entry, Q contains the orthogonal matrix used to
57 reduce the original matrix to tridiagonal form.
58 N (input) INTEGER
60 The dimension of the symmetric tridiagonal matrix. N >= 0.
61 QSIZ (input) INTEGER
63 The dimension of the orthogonal matrix used to reduce the full
64 matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
65 TLVLS (input) INTEGER
67 The total number of merging levels in the overall divide and
68 conquer tree. CURLVL (input) INTEGER The current level in the
69 overall merge routine, 0 <= CURLVL <= TLVLS. CURPBM (input)
70 INTEGER The current problem in the current level in the overall
71 merge routine (counting from upper left to lower right).
72 D (input/output) REAL array, dimension (N)
74 On entry, the eigenvalues of the rank-1-perturbed matrix. On
75 exit, the eigenvalues of the repaired matrix.
76 Q (input/output) REAL array, dimension (LDQ, N)
78 On entry, the eigenvectors of the rank-1-perturbed matrix. On
79 exit, the eigenvectors of the repaired tridiagonal matrix.
80 LDQ (input) INTEGER
82 The leading dimension of the array Q. LDQ >= max(1,N).
83 INDXQ (output) INTEGER array, dimension (N)
85 The permutation which will reintegrate the subproblem just
86 solved back into sorted order, i.e., D( INDXQ( I = 1, N ) ) will
87 be in ascending order.
88 RHO (input) REAL
90 The subdiagonal element used to create the rank-1 modification.
91 CUTPNT (input) INTEGER Contains the location of the last eigen‐
92 value in the leading sub-matrix. min(1,N) <= CUTPNT <= N.
93 QSTORE (input/output) REAL array, dimension (N**2+1) Stores
94 eigenvectors of submatrices encountered during divide and con‐
95 quer, packed together. QPTR points to beginning of the submatri‐
96 ces.
97 QPTR (input/output) INTEGER array, dimension (N+2)
99 List of indices pointing to beginning of submatrices stored in
100 QSTORE. The submatrices are numbered starting at the bottom left
101 of the divide and conquer tree, from left to right and bottom to
102 top. PRMPTR (input) INTEGER array, dimension (N lg N) Contains
103 a list of pointers which indicate where in PERM a level's permu‐
104 tation is stored. PRMPTR(i+1) - PRMPTR(i) indicates the size of
105 the permutation and also the size of the full, non-deflated
106 problem.
107 PERM (input) INTEGER array, dimension (N lg N)
109 Contains the permutations (from deflation and sorting) to be
110 applied to each eigenblock. GIVPTR (input) INTEGER array,
111 dimension (N lg N) Contains a list of pointers which indicate
112 where in GIVCOL a level's Givens rotations are stored.
113 GIVPTR(i+1) - GIVPTR(i) indicates the number of Givens rota‐
114 tions. GIVCOL (input) INTEGER array, dimension (2, N lg N) Each
115 pair of numbers indicates a pair of columns to take place in a
116 Givens rotation. GIVNUM (input) REAL array, dimension (2, N lg
117 N) Each number indicates the S value to be used in the corre‐
118 sponding Givens rotation.
119 WORK (workspace) REAL array, dimension (3*N+QSIZ*N)
121 IWORK (workspace) INTEGER array, dimension (4*N)
123 INFO (output) INTEGER
125 = 0: successful exit.
126 < 0: if INFO = -i, the i-th argument had an illegal value.
127 > 0: if INFO = 1, an eigenvalue did not converge
128
130 Based on contributions by
131 Jeff Rutter, Computer Science Division, University of California
132 at Berkeley, USA
133 LAPACK routine (version 3.2) November 2008 SLAED7(1)