1CTRSNA(1) LAPACK routine (version 3.2) CTRSNA(1)
2
6 CTRSNA - estimates reciprocal condition numbers for specified eigenval‐
7 ues and/or right eigenvectors of a complex upper triangular matrix T
8 (or of any matrix Q*T*Q**H with Q unitary)
9
11 SUBROUTINE CTRSNA( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR,
12 S, SEP, MM, M, WORK, LDWORK, RWORK, INFO )
13 CHARACTER HOWMNY, JOB
15 INTEGER INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N
17 LOGICAL SELECT( * )
19 REAL RWORK( * ), S( * ), SEP( * )
21 COMPLEX T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK(
23 LDWORK, * )
24
26 CTRSNA estimates reciprocal condition numbers for specified eigenvalues
27 and/or right eigenvectors of a complex upper triangular matrix T (or of
28 any matrix Q*T*Q**H with Q unitary).
29
31 JOB (input) CHARACTER*1
32 Specifies whether condition numbers are required for eigenval‐
33 ues (S) or eigenvectors (SEP):
34 = 'E': for eigenvalues only (S);
35 = 'V': for eigenvectors only (SEP);
36 = 'B': for both eigenvalues and eigenvectors (S and SEP).
37 HOWMNY (input) CHARACTER*1
39 = 'A': compute condition numbers for all eigenpairs;
40 = 'S': compute condition numbers for selected eigenpairs speci‐
41 fied by the array SELECT.
42 SELECT (input) LOGICAL array, dimension (N)
44 If HOWMNY = 'S', SELECT specifies the eigenpairs for which con‐
45 dition numbers are required. To select condition numbers for
46 the j-th eigenpair, SELECT(j) must be set to .TRUE.. If HOWMNY
47 = 'A', SELECT is not referenced.
48 N (input) INTEGER
50 The order of the matrix T. N >= 0.
51 T (input) COMPLEX array, dimension (LDT,N)
53 The upper triangular matrix T.
54 LDT (input) INTEGER
56 The leading dimension of the array T. LDT >= max(1,N).
57 VL (input) COMPLEX array, dimension (LDVL,M)
59 If JOB = 'E' or 'B', VL must contain left eigenvectors of T (or
60 of any Q*T*Q**H with Q unitary), corresponding to the eigen‐
61 pairs specified by HOWMNY and SELECT. The eigenvectors must be
62 stored in consecutive columns of VL, as returned by CHSEIN or
63 CTREVC. If JOB = 'V', VL is not referenced.
64 LDVL (input) INTEGER
66 The leading dimension of the array VL. LDVL >= 1; and if JOB =
67 'E' or 'B', LDVL >= N.
68 VR (input) COMPLEX array, dimension (LDVR,M)
70 If JOB = 'E' or 'B', VR must contain right eigenvectors of T
71 (or of any Q*T*Q**H with Q unitary), corresponding to the
72 eigenpairs specified by HOWMNY and SELECT. The eigenvectors
73 must be stored in consecutive columns of VR, as returned by
74 CHSEIN or CTREVC. If JOB = 'V', VR is not referenced.
75 LDVR (input) INTEGER
77 The leading dimension of the array VR. LDVR >= 1; and if JOB =
78 'E' or 'B', LDVR >= N.
79 S (output) REAL array, dimension (MM)
81 If JOB = 'E' or 'B', the reciprocal condition numbers of the
82 selected eigenvalues, stored in consecutive elements of the
83 array. Thus S(j), SEP(j), and the j-th columns of VL and VR all
84 correspond to the same eigenpair (but not in general the j-th
85 eigenpair, unless all eigenpairs are selected). If JOB = 'V',
86 S is not referenced.
87 SEP (output) REAL array, dimension (MM)
89 If JOB = 'V' or 'B', the estimated reciprocal condition numbers
90 of the selected eigenvectors, stored in consecutive elements of
91 the array. If JOB = 'E', SEP is not referenced.
92 MM (input) INTEGER
94 The number of elements in the arrays S (if JOB = 'E' or 'B')
95 and/or SEP (if JOB = 'V' or 'B'). MM >= M.
96 M (output) INTEGER
98 The number of elements of the arrays S and/or SEP actually used
99 to store the estimated condition numbers. If HOWMNY = 'A', M
100 is set to N.
101 WORK (workspace) COMPLEX array, dimension (LDWORK,N+6)
103 If JOB = 'E', WORK is not referenced.
104 LDWORK (input) INTEGER
106 The leading dimension of the array WORK. LDWORK >= 1; and if
107 JOB = 'V' or 'B', LDWORK >= N.
108 RWORK (workspace) REAL array, dimension (N)
110 If JOB = 'E', RWORK is not referenced.
111 INFO (output) INTEGER
113 = 0: successful exit
114 < 0: if INFO = -i, the i-th argument had an illegal value
115
117 The reciprocal of the condition number of an eigenvalue lambda is
118 defined as
119 S(lambda) = |v'*u| / (norm(u)*norm(v))
120 where u and v are the right and left eigenvectors of T corresponding to
121 lambda; v' denotes the conjugate transpose of v, and norm(u) denotes
122 the Euclidean norm. These reciprocal condition numbers always lie
123 between zero (very badly conditioned) and one (very well conditioned).
124 If n = 1, S(lambda) is defined to be 1.
125 An approximate error bound for a computed eigenvalue W(i) is given by
126 EPS * norm(T) / S(i)
127 where EPS is the machine precision.
128 The reciprocal of the condition number of the right eigenvector u cor‐
129 responding to lambda is defined as follows. Suppose
130 T = ( lambda c )
131 ( 0 T22 )
132 Then the reciprocal condition number is
133 SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )
134 where sigma-min denotes the smallest singular value. We approximate the
135 smallest singular value by the reciprocal of an estimate of the one-
136 norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is defined to
137 be abs(T(1,1)).
138 An approximate error bound for a computed right eigenvector VR(i) is
139 given by
140 EPS * norm(T) / SEP(i)
141 LAPACK routine (version 3.2) November 2008 CTRSNA(1)