1SLAED1(1) LAPACK routine (version 3.2) SLAED1(1)
2
6 SLAED1 - computes the updated eigensystem of a diagonal matrix after
7 modification by a rank-one symmetric matrix
8
10 SUBROUTINE SLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK, INFO
11 )
12 INTEGER CUTPNT, INFO, LDQ, N
14 REAL RHO
16 INTEGER INDXQ( * ), IWORK( * )
18 REAL D( * ), Q( LDQ, * ), WORK( * )
20
22 SLAED1 computes the updated eigensystem of a diagonal matrix after mod‐
23 ification by a rank-one symmetric matrix. This routine is used only
24 for the eigenproblem which requires all eigenvalues and eigenvectors of
25 a tridiagonal matrix. SLAED7 handles the case in which eigenvalues
26 only or eigenvalues and eigenvectors of a full symmetric matrix (which
27 was reduced to tridiagonal form) are desired.
28 T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out)
29 where Z = Q'u, u is a vector of length N with ones in the
30 CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
31 The eigenvectors of the original matrix are stored in Q, and the
32 eigenvalues are in D. The algorithm consists of three stages:
33 The first stage consists of deflating the size of the problem
34 when there are multiple eigenvalues or if there is a zero in
35 the Z vector. For each such occurence the dimension of the
36 secular equation problem is reduced by one. This stage is
37 performed by the routine SLAED2.
38 The second stage consists of calculating the updated
39 eigenvalues. This is done by finding the roots of the secular
40 equation via the routine SLAED4 (as called by SLAED3).
41 This routine also calculates the eigenvectors of the current
42 problem.
43 The final stage consists of computing the updated eigenvectors
44 directly using the updated eigenvalues. The eigenvectors for
45 the current problem are multiplied with the eigenvectors from
46 the overall problem.
47
49 N (input) INTEGER
50 The dimension of the symmetric tridiagonal matrix. N >= 0.
51 D (input/output) REAL array, dimension (N)
53 On entry, the eigenvalues of the rank-1-perturbed matrix. On
54 exit, the eigenvalues of the repaired matrix.
55 Q (input/output) REAL array, dimension (LDQ,N)
57 On entry, the eigenvectors of the rank-1-perturbed matrix. On
58 exit, the eigenvectors of the repaired tridiagonal matrix.
59 LDQ (input) INTEGER
61 The leading dimension of the array Q. LDQ >= max(1,N).
62 INDXQ (input/output) INTEGER array, dimension (N)
64 On entry, the permutation which separately sorts the two sub‐
65 problems in D into ascending order. On exit, the permutation
66 which will reintegrate the subproblems back into sorted order,
67 i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.
68 RHO (input) REAL
70 The subdiagonal entry used to create the rank-1 modification.
71 CUTPNT (input) INTEGER The location of the last eigenvalue in
72 the leading sub-matrix. min(1,N) <= CUTPNT <= N/2.
73 WORK (workspace) REAL array, dimension (4*N + N**2)
75 IWORK (workspace) INTEGER array, dimension (4*N)
77 INFO (output) INTEGER
79 = 0: successful exit.
80 < 0: if INFO = -i, the i-th argument had an illegal value.
81 > 0: if INFO = 1, an eigenvalue did not converge
82
84 Based on contributions by
85 Jeff Rutter, Computer Science Division, University of California
86 at Berkeley, USA
87 Modified by Francoise Tisseur, University of Tennessee.
88 LAPACK routine (version 3.2) November 2008 SLAED1(1)