1CLAED7(1) LAPACK routine (version 3.2) CLAED7(1)
2
6 CLAED7 - computes the updated eigensystem of a diagonal matrix after
7 modification by a rank-one symmetric matrix
8
10 SUBROUTINE CLAED7( 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 REAL RHO
17 INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ), IWORK( * ),
19 PERM( * ), PRMPTR( * ), QPTR( * )
20 REAL D( * ), GIVNUM( 2, * ), QSTORE( * ), RWORK( * )
22 COMPLEX Q( LDQ, * ), WORK( * )
24
26 CLAED7 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 or banded Hermitian matrix that has been reduced to
30 tridiagonal form.
31 T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out)
32 where Z = Q'u, u is a vector of length N with ones in the
33 CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
34 The eigenvectors of the original matrix are stored in Q, and the
35 eigenvalues are in D. The algorithm consists of three stages:
36 The first stage consists of deflating the size of the problem
37 when there are multiple eigenvalues or if there is a zero in
38 the Z vector. For each such occurence the dimension of the
39 secular equation problem is reduced by one. This stage is
40 performed by the routine SLAED2.
41 The second stage consists of calculating the updated
42 eigenvalues. This is done by finding the roots of the secular
43 equation via the routine SLAED4 (as called by SLAED3).
44 This routine also calculates the eigenvectors of the current
45 problem.
46 The final stage consists of computing the updated eigenvectors
47 directly using the updated eigenvalues. The eigenvectors for
48 the current problem are multiplied with the eigenvectors from
49 the overall problem.
50
52 N (input) INTEGER
53 The dimension of the symmetric tridiagonal matrix. N >= 0.
54 CUTPNT (input) INTEGER Contains the location of the last eigen‐
55 value in the leading sub-matrix. min(1,N) <= CUTPNT <= N.
56 QSIZ (input) INTEGER
58 The dimension of the unitary matrix used to reduce the full
59 matrix to tridiagonal form. QSIZ >= N.
60 TLVLS (input) INTEGER
62 The total number of merging levels in the overall divide and
63 conquer tree. CURLVL (input) INTEGER The current level in the
64 overall merge routine, 0 <= curlvl <= tlvls. CURPBM (input)
65 INTEGER The current problem in the current level in the overall
66 merge routine (counting from upper left to lower right).
67 D (input/output) REAL array, dimension (N)
69 On entry, the eigenvalues of the rank-1-perturbed matrix. On
70 exit, the eigenvalues of the repaired matrix.
71 Q (input/output) COMPLEX array, dimension (LDQ,N)
73 On entry, the eigenvectors of the rank-1-perturbed matrix. On
74 exit, the eigenvectors of the repaired tridiagonal matrix.
75 LDQ (input) INTEGER
77 The leading dimension of the array Q. LDQ >= max(1,N).
78 RHO (input) REAL
80 Contains the subdiagonal element used to create the rank-1 modi‐
81 fication.
82 INDXQ (output) INTEGER array, dimension (N)
84 This contains the permutation which will reintegrate the sub‐
85 problem just solved back into sorted order, ie. D( INDXQ( I = 1,
86 N ) ) will be in ascending order.
87 IWORK (workspace) INTEGER array, dimension (4*N)
89 RWORK (workspace) REAL array,
91 dimension (3*N+2*QSIZ*N)
92 WORK (workspace) COMPLEX array, dimension (QSIZ*N)
94 QSTORE (input/output) REAL array, dimension (N**2+1) Stores
95 eigenvectors of submatrices encountered during divide and con‐
96 quer, packed together. QPTR points to beginning of the submatri‐
97 ces.
98 QPTR (input/output) INTEGER array, dimension (N+2)
100 List of indices pointing to beginning of submatrices stored in
101 QSTORE. The submatrices are numbered starting at the bottom left
102 of the divide and conquer tree, from left to right and bottom to
103 top. PRMPTR (input) INTEGER array, dimension (N lg N) Contains
104 a list of pointers which indicate where in PERM a level's permu‐
105 tation is stored. PRMPTR(i+1) - PRMPTR(i) indicates the size of
106 the permutation and also the size of the full, non-deflated
107 problem.
108 PERM (input) INTEGER array, dimension (N lg N)
110 Contains the permutations (from deflation and sorting) to be
111 applied to each eigenblock. GIVPTR (input) INTEGER array,
112 dimension (N lg N) Contains a list of pointers which indicate
113 where in GIVCOL a level's Givens rotations are stored.
114 GIVPTR(i+1) - GIVPTR(i) indicates the number of Givens rota‐
115 tions. GIVCOL (input) INTEGER array, dimension (2, N lg N) Each
116 pair of numbers indicates a pair of columns to take place in a
117 Givens rotation. GIVNUM (input) REAL array, dimension (2, N lg
118 N) Each number indicates the S value to be used in the corre‐
119 sponding Givens rotation.
120 INFO (output) INTEGER
122 = 0: successful exit.
123 < 0: if INFO = -i, the i-th argument had an illegal value.
124 > 0: if INFO = 1, an eigenvalue did not converge
125 LAPACK routine (version 3.2) November 2008 CLAED7(1)