1DLAED1(1) LAPACK routine (version 3.1) DLAED1(1)
2
6 DLAED1 - the updated eigensystem of a diagonal matrix after modifica‐
7 tion by a rank-one symmetric matrix
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10 SUBROUTINE DLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK, INFO
11 )
12 INTEGER CUTPNT, INFO, LDQ, N
14 DOUBLE PRECISION RHO
16 INTEGER INDXQ( * ), IWORK( * )
18 DOUBLE PRECISION D( * ), Q( LDQ, * ), WORK( * )
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22 DLAED1 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. DLAED7 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)
30 where Z = Q'u, u is a vector of length N with ones in the
32 CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
33 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
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 DLAED2.
42 The second stage consists of calculating the updated
44 eigenvalues. This is done by finding the roots of the secular
45 equation via the routine DLAED4 (as called by DLAED3).
46 This routine also calculates the eigenvectors of the current
47 problem.
48 The final stage consists of computing the updated eigenvectors
50 directly using the updated eigenvalues. The eigenvectors for
51 the current problem are multiplied with the eigenvectors from
52 the overall problem.
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56 N (input) INTEGER
57 The dimension of the symmetric tridiagonal matrix. N >= 0.
58 D (input/output) DOUBLE PRECISION array, dimension (N)
60 On entry, the eigenvalues of the rank-1-perturbed matrix. On
61 exit, the eigenvalues of the repaired matrix.
62 Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
64 On entry, the eigenvectors of the rank-1-perturbed matrix. On
65 exit, the eigenvectors of the repaired tridiagonal matrix.
66 LDQ (input) INTEGER
68 The leading dimension of the array Q. LDQ >= max(1,N).
69 INDXQ (input/output) INTEGER array, dimension (N)
71 On entry, the permutation which separately sorts the two sub‐
72 problems in D into ascending order. On exit, the permutation
73 which will reintegrate the subproblems back into sorted order,
74 i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.
75 RHO (input) DOUBLE PRECISION
77 The subdiagonal entry used to create the rank-1 modification.
78 CUTPNT (input) INTEGER The location of the last eigenvalue in
80 the leading sub-matrix. min(1,N) <= CUTPNT <= N/2.
81 WORK (workspace) DOUBLE PRECISION array, dimension (4*N + N**2)
83 IWORK (workspace) INTEGER array, dimension (4*N)
85 INFO (output) INTEGER
87 = 0: successful exit.
88 < 0: if INFO = -i, the i-th argument had an illegal value.
89 > 0: if INFO = 1, an eigenvalue did not converge
90
92 Based on contributions by
93 Jeff Rutter, Computer Science Division, University of California
94 at Berkeley, USA
95 Modified by Francoise Tisseur, University of Tennessee.
96 LAPACK routine (version 3.1) November 2006 DLAED1(1)