1SLAED7(1) LAPACK routine (version 3.1) SLAED7(1)
2
6 SLAED7 - the updated eigensystem of a diagonal matrix after modifica‐
7 tion 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)
34 where Z = Q'u, u is a vector of length N with ones in the
36 CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
37 The eigenvectors of the original matrix are stored in Q, and the
39 eigenvalues are in D. The algorithm consists of three stages:
40 The first stage consists of deflating the size of the problem
42 when there are multiple eigenvalues or if there is a zero in
43 the Z vector. For each such occurence the dimension of the
44 secular equation problem is reduced by one. This stage is
45 performed by the routine SLAED8.
46 The second stage consists of calculating the updated
48 eigenvalues. This is done by finding the roots of the secular
49 equation via the routine SLAED4 (as called by SLAED9).
50 This routine also calculates the eigenvectors of the current
51 problem.
52 The final stage consists of computing the updated eigenvectors
54 directly using the updated eigenvalues. The eigenvectors for
55 the current problem are multiplied with the eigenvectors from
56 the overall problem.
57
60 ICOMPQ (input) INTEGER
61 = 0: Compute eigenvalues only.
62 = 1: Compute eigenvectors of original dense symmetric matrix
63 also. On entry, Q contains the orthogonal matrix used to
64 reduce the original matrix to tridiagonal form.
65 N (input) INTEGER
67 The dimension of the symmetric tridiagonal matrix. N >= 0.
68 QSIZ (input) INTEGER
70 The dimension of the orthogonal matrix used to reduce the full
71 matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
72 TLVLS (input) INTEGER
74 The total number of merging levels in the overall divide and
75 conquer tree.
76 CURLVL (input) INTEGER The current level in the overall merge
78 routine, 0 <= CURLVL <= TLVLS.
79 CURPBM (input) INTEGER The current problem in the current level
81 in the overall merge routine (counting from upper left to lower
82 right).
83 D (input/output) REAL array, dimension (N)
85 On entry, the eigenvalues of the rank-1-perturbed matrix. On
86 exit, the eigenvalues of the repaired matrix.
87 Q (input/output) REAL array, dimension (LDQ, N)
89 On entry, the eigenvectors of the rank-1-perturbed matrix. On
90 exit, the eigenvectors of the repaired tridiagonal matrix.
91 LDQ (input) INTEGER
93 The leading dimension of the array Q. LDQ >= max(1,N).
94 INDXQ (output) INTEGER array, dimension (N)
96 The permutation which will reintegrate the subproblem just
97 solved back into sorted order, i.e., D( INDXQ( I = 1, N ) ) will
98 be in ascending order.
99 RHO (input) REAL
101 The subdiagonal element used to create the rank-1 modification.
102 CUTPNT (input) INTEGER Contains the location of the last eigen‐
104 value in the leading sub-matrix. min(1,N) <= CUTPNT <= N.
105 QSTORE (input/output) REAL array, dimension (N**2+1) Stores
107 eigenvectors of submatrices encountered during divide and con‐
108 quer, packed together. QPTR points to beginning of the submatri‐
109 ces.
110 QPTR (input/output) INTEGER array, dimension (N+2)
112 List of indices pointing to beginning of submatrices stored in
113 QSTORE. The submatrices are numbered starting at the bottom left
114 of the divide and conquer tree, from left to right and bottom to
115 top.
116 PRMPTR (input) INTEGER array, dimension (N lg N) Contains a list
118 of pointers which indicate where in PERM a level's permutation
119 is stored. PRMPTR(i+1) - PRMPTR(i) indicates the size of the
120 permutation and also the size of the full, non-deflated problem.
121 PERM (input) INTEGER array, dimension (N lg N)
123 Contains the permutations (from deflation and sorting) to be
124 applied to each eigenblock.
125 GIVPTR (input) INTEGER array, dimension (N lg N) Contains a list
127 of pointers which indicate where in GIVCOL a level's Givens
128 rotations are stored. GIVPTR(i+1) - GIVPTR(i) indicates the
129 number of Givens rotations.
130 GIVCOL (input) INTEGER array, dimension (2, N lg N) Each pair of
132 numbers indicates a pair of columns to take place in a Givens
133 rotation.
134 GIVNUM (input) REAL array, dimension (2, N lg N) Each number
136 indicates the S value to be used in the corresponding Givens
137 rotation.
138 WORK (workspace) REAL array, dimension (3*N+QSIZ*N)
140 IWORK (workspace) INTEGER array, dimension (4*N)
142 INFO (output) INTEGER
144 = 0: successful exit.
145 < 0: if INFO = -i, the i-th argument had an illegal value.
146 > 0: if INFO = 1, an eigenvalue did not converge
147
149 Based on contributions by
150 Jeff Rutter, Computer Science Division, University of California
151 at Berkeley, USA
152 LAPACK routine (version 3.1) November 2006 SLAED7(1)