1SLAED8(1) LAPACK routine (version 3.2) SLAED8(1)
2
6 SLAED8 - merges the two sets of eigenvalues together into a single
7 sorted set
8
10 SUBROUTINE SLAED8( 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 REAL RHO
17 INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ), INDXQ( * ),
19 PERM( * )
20 REAL D( * ), DLAMDA( * ), GIVNUM( 2, * ), Q( LDQ, * ),
22 Q2( LDQ2, * ), W( * ), Z( * )
23
25 SLAED8 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) REAL 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) REAL 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) REAL
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 SLAED3. CUTPNT (input) INTEGER The location of the
77 last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT
78 <= N.
79 Z (input) REAL 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) REAL array,
85 dimension (N) A copy of the first K eigenvalues which will be
86 used by SLAED3 to form the secular equation.
87 Q2 (output) REAL 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 SLAED7 in a matrix
91 multiply (SGEMM) to update the new eigenvectors.
92 LDQ2 (input) INTEGER
94 The leading dimension of the array Q2. LDQ2 >= max(1,N).
95 W (output) REAL array, dimension (N)
97 The first k values of the final deflation-altered z-vector and
98 will be passed to SLAED3.
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) REAL array, dimension (2, N) Each number indicates the
107 S value to be used in the corresponding Givens rotation.
108 INDXP (workspace) INTEGER array, dimension (N)
110 The permutation used to place deflated values of D at the end of
111 the array. INDXP(1:K) points to the nondeflated D-values
112 and INDXP(K+1:N) points to the deflated eigenvalues.
113 INDX (workspace) INTEGER array, dimension (N)
115 The permutation used to sort the contents of D into ascending
116 order.
117 INFO (output) INTEGER
119 = 0: successful exit.
120 < 0: if INFO = -i, the i-th argument had an illegal value.
121
123 Based on contributions by
124 Jeff Rutter, Computer Science Division, University of California
125 at Berkeley, USA
126 LAPACK routine (version 3.2) November 2008 SLAED8(1)