1SLAED8(1) LAPACK routine (version 3.1) SLAED8(1)
2
6 SLAED8 - the two sets of eigenvalues together into a single sorted set
7
9 SUBROUTINE SLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, CUTPNT,
10 Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR, GIVCOL,
11 GIVNUM, INDXP, INDX, INFO )
12 INTEGER CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N, QSIZ
14 REAL RHO
16 INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ), INDXQ( * ),
18 PERM( * )
19 REAL D( * ), DLAMDA( * ), GIVNUM( 2, * ), Q( LDQ, * ),
21 Q2( LDQ2, * ), W( * ), Z( * )
22
24 SLAED8 merges the two sets of eigenvalues together into a single sorted
25 set. Then it tries to deflate the size of the problem. There are two
26 ways in which deflation can occur: when two or more eigenvalues are
27 close together or if there is a tiny element in the Z vector. For each
28 such occurrence the order of the related secular equation problem is
29 reduced by one.
30
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.
77 CUTPNT (input) INTEGER The location of the last eigenvalue in
79 the leading sub-matrix. min(1,N) <= CUTPNT <= N.
80 Z (input) REAL array, dimension (N)
82 On entry, Z contains the updating vector (the last row of the
83 first sub-eigenvector matrix and the first row of the second
84 sub-eigenvector matrix). On exit, the contents of Z are
85 destroyed by the updating process.
86 DLAMDA (output) REAL array, dimension (N) A copy of the first K
88 eigenvalues which will be used by SLAED3 to form the secular
89 equation.
90 Q2 (output) REAL array, dimension (LDQ2,N)
92 If ICOMPQ = 0, Q2 is not referenced. Otherwise, a copy of the
93 first K eigenvectors which will be used by SLAED7 in a matrix
94 multiply (SGEMM) to update the new eigenvectors.
95 LDQ2 (input) INTEGER
97 The leading dimension of the array Q2. LDQ2 >= max(1,N).
98 W (output) REAL array, dimension (N)
100 The first k values of the final deflation-altered z-vector and
101 will be passed to SLAED3.
102 PERM (output) INTEGER array, dimension (N)
104 The permutations (from deflation and sorting) to be applied to
105 each eigenblock.
106 GIVPTR (output) INTEGER The number of Givens rotations which
108 took place in this subproblem.
109 GIVCOL (output) INTEGER array, dimension (2, N) Each pair of
111 numbers indicates a pair of columns to take place in a Givens
112 rotation.
113 GIVNUM (output) REAL array, dimension (2, N) Each number indi‐
115 cates the S value to be used in the corresponding Givens rota‐
116 tion.
117 INDXP (workspace) INTEGER array, dimension (N)
119 The permutation used to place deflated values of D at the end of
120 the array. INDXP(1:K) points to the nondeflated D-values
121 and INDXP(K+1:N) points to the deflated eigenvalues.
122 INDX (workspace) INTEGER array, dimension (N)
124 The permutation used to sort the contents of D into ascending
125 order.
126 INFO (output) INTEGER
128 = 0: successful exit.
129 < 0: if INFO = -i, the i-th argument had an illegal value.
130
132 Based on contributions by
133 Jeff Rutter, Computer Science Division, University of California
134 at Berkeley, USA
135 LAPACK routine (version 3.1) November 2006 SLAED8(1)