1DTRSNA(1) LAPACK routine (version 3.1) DTRSNA(1)
2
6 DTRSNA - reciprocal condition numbers for specified eigenvalues and/or
7 right eigenvectors of a real upper quasi-triangular matrix T (or of any
8 matrix Q*T*Q**T with Q orthogonal)
9
11 SUBROUTINE DTRSNA( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR,
12 S, SEP, MM, M, WORK, LDWORK, IWORK, INFO )
13 CHARACTER HOWMNY, JOB
15 INTEGER INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N
17 LOGICAL SELECT( * )
19 INTEGER IWORK( * )
21 DOUBLE PRECISION S( * ), SEP( * ), T( LDT, * ), VL( LDVL, *
23 ), VR( LDVR, * ), WORK( LDWORK, * )
24
26 DTRSNA estimates reciprocal condition numbers for specified eigenvalues
27 and/or right eigenvectors of a real upper quasi-triangular matrix T (or
28 of any matrix Q*T*Q**T with Q orthogonal).
29 T must be in Schur canonical form (as returned by DHSEQR), that is,
31 block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
32 2-by-2 diagonal block has its diagonal elements equal and its off-diag‐
33 onal elements of opposite sign.
34
37 JOB (input) CHARACTER*1
38 Specifies whether condition numbers are required for eigenval‐
39 ues (S) or eigenvectors (SEP):
40 = 'E': for eigenvalues only (S);
41 = 'V': for eigenvectors only (SEP);
42 = 'B': for both eigenvalues and eigenvectors (S and SEP).
43 HOWMNY (input) CHARACTER*1
45 = 'A': compute condition numbers for all eigenpairs;
46 = 'S': compute condition numbers for selected eigenpairs speci‐
47 fied by the array SELECT.
48 SELECT (input) LOGICAL array, dimension (N)
50 If HOWMNY = 'S', SELECT specifies the eigenpairs for which con‐
51 dition numbers are required. To select condition numbers for
52 the eigenpair corresponding to a real eigenvalue w(j),
53 SELECT(j) must be set to .TRUE.. To select condition numbers
54 corresponding to a complex conjugate pair of eigenvalues w(j)
55 and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
56 set to .TRUE.. If HOWMNY = 'A', SELECT is not referenced.
57 N (input) INTEGER
59 The order of the matrix T. N >= 0.
60 T (input) DOUBLE PRECISION array, dimension (LDT,N)
62 The upper quasi-triangular matrix T, in Schur canonical form.
63 LDT (input) INTEGER
65 The leading dimension of the array T. LDT >= max(1,N).
66 VL (input) DOUBLE PRECISION array, dimension (LDVL,M)
68 If JOB = 'E' or 'B', VL must contain left eigenvectors of T (or
69 of any Q*T*Q**T with Q orthogonal), corresponding to the eigen‐
70 pairs specified by HOWMNY and SELECT. The eigenvectors must be
71 stored in consecutive columns of VL, as returned by DHSEIN or
72 DTREVC. If JOB = 'V', VL is not referenced.
73 LDVL (input) INTEGER
75 The leading dimension of the array VL. LDVL >= 1; and if JOB =
76 'E' or 'B', LDVL >= N.
77 VR (input) DOUBLE PRECISION array, dimension (LDVR,M)
79 If JOB = 'E' or 'B', VR must contain right eigenvectors of T
80 (or of any Q*T*Q**T with Q orthogonal), corresponding to the
81 eigenpairs specified by HOWMNY and SELECT. The eigenvectors
82 must be stored in consecutive columns of VR, as returned by
83 DHSEIN or DTREVC. If JOB = 'V', VR is not referenced.
84 LDVR (input) INTEGER
86 The leading dimension of the array VR. LDVR >= 1; and if JOB =
87 'E' or 'B', LDVR >= N.
88 S (output) DOUBLE PRECISION array, dimension (MM)
90 If JOB = 'E' or 'B', the reciprocal condition numbers of the
91 selected eigenvalues, stored in consecutive elements of the
92 array. For a complex conjugate pair of eigenvalues two consecu‐
93 tive elements of S are set to the same value. Thus S(j),
94 SEP(j), and the j-th columns of VL and VR all correspond to the
95 same eigenpair (but not in general the j-th eigenpair, unless
96 all eigenpairs are selected). If JOB = 'V', S is not refer‐
97 enced.
98 SEP (output) DOUBLE PRECISION array, dimension (MM)
100 If JOB = 'V' or 'B', the estimated reciprocal condition numbers
101 of the selected eigenvectors, stored in consecutive elements of
102 the array. For a complex eigenvector two consecutive elements
103 of SEP are set to the same value. If the eigenvalues cannot be
104 reordered to compute SEP(j), SEP(j) is set to 0; this can only
105 occur when the true value would be very small anyway. If JOB =
106 'E', SEP is not referenced.
107 MM (input) INTEGER
109 The number of elements in the arrays S (if JOB = 'E' or 'B')
110 and/or SEP (if JOB = 'V' or 'B'). MM >= M.
111 M (output) INTEGER
113 The number of elements of the arrays S and/or SEP actually used
114 to store the estimated condition numbers. If HOWMNY = 'A', M
115 is set to N.
116 WORK (workspace) DOUBLE PRECISION array, dimension (LDWORK,N+6)
118 If JOB = 'E', WORK is not referenced.
119 LDWORK (input) INTEGER
121 The leading dimension of the array WORK. LDWORK >= 1; and if
122 JOB = 'V' or 'B', LDWORK >= N.
123 IWORK (workspace) INTEGER array, dimension (2*(N-1))
125 If JOB = 'E', IWORK is not referenced.
126 INFO (output) INTEGER
128 = 0: successful exit
129 < 0: if INFO = -i, the i-th argument had an illegal value
130
132 The reciprocal of the condition number of an eigenvalue lambda is
133 defined as
134 S(lambda) = |v'*u| / (norm(u)*norm(v))
136 where u and v are the right and left eigenvectors of T corresponding to
138 lambda; v' denotes the conjugate-transpose of v, and norm(u) denotes
139 the Euclidean norm. These reciprocal condition numbers always lie
140 between zero (very badly conditioned) and one (very well conditioned).
141 If n = 1, S(lambda) is defined to be 1.
142 An approximate error bound for a computed eigenvalue W(i) is given by
144 EPS * norm(T) / S(i)
146 where EPS is the machine precision.
148 The reciprocal of the condition number of the right eigenvector u cor‐
150 responding to lambda is defined as follows. Suppose
151 T = ( lambda c )
153 ( 0 T22 )
154 Then the reciprocal condition number is
156 SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )
158 where sigma-min denotes the smallest singular value. We approximate the
160 smallest singular value by the reciprocal of an estimate of the one-
161 norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is defined to
162 be abs(T(1,1)).
163 An approximate error bound for a computed right eigenvector VR(i) is
165 given by
166 EPS * norm(T) / SEP(i)
168 LAPACK routine (version 3.1) November 2006 DTRSNA(1)