1CLAED7(1) LAPACK routine (version 3.1) CLAED7(1)
2
6 CLAED7 - the updated eigensystem of a diagonal matrix after modifica‐
7 tion 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)
33 where Z = Q'u, u is a vector of length N with ones in the
35 CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
36 The eigenvectors of the original matrix are stored in Q, and the
38 eigenvalues are in D. The algorithm consists of three stages:
39 The first stage consists of deflating the size of the problem
41 when there are multiple eigenvalues or if there is a zero in
42 the Z vector. For each such occurence the dimension of the
43 secular equation problem is reduced by one. This stage is
44 performed by the routine SLAED2.
45 The second stage consists of calculating the updated
47 eigenvalues. This is done by finding the roots of the secular
48 equation via the routine SLAED4 (as called by SLAED3).
49 This routine also calculates the eigenvectors of the current
50 problem.
51 The final stage consists of computing the updated eigenvectors
53 directly using the updated eigenvalues. The eigenvectors for
54 the current problem are multiplied with the eigenvectors from
55 the overall problem.
56
59 N (input) INTEGER
60 The dimension of the symmetric tridiagonal matrix. N >= 0.
61 CUTPNT (input) INTEGER Contains the location of the last eigen‐
63 value in the leading sub-matrix. min(1,N) <= CUTPNT <= N.
64 QSIZ (input) INTEGER
66 The dimension of the unitary matrix used to reduce the full
67 matrix to tridiagonal form. QSIZ >= N.
68 TLVLS (input) INTEGER
70 The total number of merging levels in the overall divide and
71 conquer tree.
72 CURLVL (input) INTEGER The current level in the overall merge
74 routine, 0 <= curlvl <= tlvls.
75 CURPBM (input) INTEGER The current problem in the current level
77 in the overall merge routine (counting from upper left to lower
78 right).
79 D (input/output) REAL array, dimension (N)
81 On entry, the eigenvalues of the rank-1-perturbed matrix. On
82 exit, the eigenvalues of the repaired matrix.
83 Q (input/output) COMPLEX array, dimension (LDQ,N)
85 On entry, the eigenvectors of the rank-1-perturbed matrix. On
86 exit, the eigenvectors of the repaired tridiagonal matrix.
87 LDQ (input) INTEGER
89 The leading dimension of the array Q. LDQ >= max(1,N).
90 RHO (input) REAL
92 Contains the subdiagonal element used to create the rank-1 modi‐
93 fication.
94 INDXQ (output) INTEGER array, dimension (N)
96 This contains the permutation which will reintegrate the sub‐
97 problem just solved back into sorted order, ie. D( INDXQ( I = 1,
98 N ) ) will be in ascending order.
99 IWORK (workspace) INTEGER array, dimension (4*N)
101 RWORK (workspace) REAL array,
103 dimension (3*N+2*QSIZ*N)
104 WORK (workspace) COMPLEX array, dimension (QSIZ*N)
106 QSTORE (input/output) REAL array, dimension (N**2+1) Stores
108 eigenvectors of submatrices encountered during divide and con‐
109 quer, packed together. QPTR points to beginning of the submatri‐
110 ces.
111 QPTR (input/output) INTEGER array, dimension (N+2)
113 List of indices pointing to beginning of submatrices stored in
114 QSTORE. The submatrices are numbered starting at the bottom left
115 of the divide and conquer tree, from left to right and bottom to
116 top.
117 PRMPTR (input) INTEGER array, dimension (N lg N) Contains a list
119 of pointers which indicate where in PERM a level's permutation
120 is stored. PRMPTR(i+1) - PRMPTR(i) indicates the size of the
121 permutation and also the size of the full, non-deflated problem.
122 PERM (input) INTEGER array, dimension (N lg N)
124 Contains the permutations (from deflation and sorting) to be
125 applied to each eigenblock.
126 GIVPTR (input) INTEGER array, dimension (N lg N) Contains a list
128 of pointers which indicate where in GIVCOL a level's Givens
129 rotations are stored. GIVPTR(i+1) - GIVPTR(i) indicates the
130 number of Givens rotations.
131 GIVCOL (input) INTEGER array, dimension (2, N lg N) Each pair of
133 numbers indicates a pair of columns to take place in a Givens
134 rotation.
135 GIVNUM (input) REAL array, dimension (2, N lg N) Each number
137 indicates the S value to be used in the corresponding Givens
138 rotation.
139 INFO (output) INTEGER
141 = 0: successful exit.
142 < 0: if INFO = -i, the i-th argument had an illegal value.
143 > 0: if INFO = 1, an eigenvalue did not converge
144 LAPACK routine (version 3.1) November 2006 CLAED7(1)