1DLAED8(1) LAPACK routine (version 3.2) DLAED8(1)
2
6 DLAED8 - merges the two sets of eigenvalues together into a single
7 sorted set
8
10 SUBROUTINE DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, CUTPNT,
11 Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR, GIVCOL,
12 GIVNUM, INDXP, INDX, INFO )
13 INTEGER CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N, QSIZ
15 DOUBLE PRECISION RHO
17 INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ), INDXQ( * ),
19 PERM( * )
20 DOUBLE PRECISION D( * ), DLAMDA( * ), GIVNUM( 2, * ), Q(
22 LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * )
23
25 DLAED8 merges the two sets of eigenvalues together into a single sorted
26 set. Then it tries to deflate the size of the problem. There are two
27 ways in which deflation can occur: when two or more eigenvalues are
28 close together or if there is a tiny element in the Z vector. For each
29 such occurrence the order of the related secular equation problem is
30 reduced by one.
31
33 ICOMPQ (input) INTEGER
34 = 0: Compute eigenvalues only.
35 = 1: Compute eigenvectors of original dense symmetric matrix
36 also. On entry, Q contains the orthogonal matrix used to
37 reduce the original matrix to tridiagonal form.
38 K (output) INTEGER
40 The number of non-deflated eigenvalues, and the order of the
41 related secular equation.
42 N (input) INTEGER
44 The dimension of the symmetric tridiagonal matrix. N >= 0.
45 QSIZ (input) INTEGER
47 The dimension of the orthogonal matrix used to reduce the full
48 matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
49 D (input/output) DOUBLE PRECISION array, dimension (N)
51 On entry, the eigenvalues of the two submatrices to be combined.
52 On exit, the trailing (N-K) updated eigenvalues (those which
53 were deflated) sorted into increasing order.
54 Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
56 If ICOMPQ = 0, Q is not referenced. Otherwise, on entry, Q con‐
57 tains the eigenvectors of the partially solved system which has
58 been previously updated in matrix multiplies with other par‐
59 tially solved eigensystems. On exit, Q contains the trailing
60 (N-K) updated eigenvectors (those which were deflated) in its
61 last N-K columns.
62 LDQ (input) INTEGER
64 The leading dimension of the array Q. LDQ >= max(1,N).
65 INDXQ (input) INTEGER array, dimension (N)
67 The permutation which separately sorts the two sub-problems in D
68 into ascending order. Note that elements in the second half of
69 this permutation must first have CUTPNT added to their values in
70 order to be accurate.
71 RHO (input/output) DOUBLE PRECISION
73 On entry, the off-diagonal element associated with the rank-1
74 cut which originally split the two submatrices which are now
75 being recombined. On exit, RHO has been modified to the value
76 required by DLAED3. CUTPNT (input) INTEGER The location of the
77 last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT
78 <= N.
79 Z (input) DOUBLE PRECISION array, dimension (N)
81 On entry, Z contains the updating vector (the last row of the
82 first sub-eigenvector matrix and the first row of the second
83 sub-eigenvector matrix). On exit, the contents of Z are
84 destroyed by the updating process. DLAMDA (output) DOUBLE PRE‐
85 CISION array, dimension (N) A copy of the first K eigenvalues
86 which will be used by DLAED3 to form the secular equation.
87 Q2 (output) DOUBLE PRECISION array, dimension (LDQ2,N)
89 If ICOMPQ = 0, Q2 is not referenced. Otherwise, a copy of the
90 first K eigenvectors which will be used by DLAED7 in a matrix
91 multiply (DGEMM) to update the new eigenvectors.
92 LDQ2 (input) INTEGER
94 The leading dimension of the array Q2. LDQ2 >= max(1,N).
95 W (output) DOUBLE PRECISION array, dimension (N)
97 The first k values of the final deflation-altered z-vector and
98 will be passed to DLAED3.
99 PERM (output) INTEGER array, dimension (N)
101 The permutations (from deflation and sorting) to be applied to
102 each eigenblock. GIVPTR (output) INTEGER The number of Givens
103 rotations which took place in this subproblem. GIVCOL (output)
104 INTEGER array, dimension (2, N) Each pair of numbers indicates a
105 pair of columns to take place in a Givens rotation. GIVNUM
106 (output) DOUBLE PRECISION array, dimension (2, N) Each number
107 indicates the S value to be used in the corresponding Givens
108 rotation.
109 INDXP (workspace) INTEGER array, dimension (N)
111 The permutation used to place deflated values of D at the end of
112 the array. INDXP(1:K) points to the nondeflated D-values
113 and INDXP(K+1:N) points to the deflated eigenvalues.
114 INDX (workspace) INTEGER array, dimension (N)
116 The permutation used to sort the contents of D into ascending
117 order.
118 INFO (output) INTEGER
120 = 0: successful exit.
121 < 0: if INFO = -i, the i-th argument had an illegal value.
122
124 Based on contributions by
125 Jeff Rutter, Computer Science Division, University of California
126 at Berkeley, USA
127 LAPACK routine (version 3.2) November 2008 DLAED8(1)