1CTRSNA(1) LAPACK routine (version 3.1) CTRSNA(1)
2
6 CTRSNA - reciprocal condition numbers for specified eigenvalues and/or
7 right eigenvectors of a complex upper triangular matrix T (or of any
8 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
32 JOB (input) CHARACTER*1
33 Specifies whether condition numbers are required for eigenval‐
34 ues (S) or eigenvectors (SEP):
35 = 'E': for eigenvalues only (S);
36 = 'V': for eigenvectors only (SEP);
37 = 'B': for both eigenvalues and eigenvectors (S and SEP).
38 HOWMNY (input) CHARACTER*1
40 = 'A': compute condition numbers for all eigenpairs;
41 = 'S': compute condition numbers for selected eigenpairs speci‐
42 fied by the array SELECT.
43 SELECT (input) LOGICAL array, dimension (N)
45 If HOWMNY = 'S', SELECT specifies the eigenpairs for which con‐
46 dition numbers are required. To select condition numbers for
47 the j-th eigenpair, SELECT(j) must be set to .TRUE.. If HOWMNY
48 = 'A', SELECT is not referenced.
49 N (input) INTEGER
51 The order of the matrix T. N >= 0.
52 T (input) COMPLEX array, dimension (LDT,N)
54 The upper triangular matrix T.
55 LDT (input) INTEGER
57 The leading dimension of the array T. LDT >= max(1,N).
58 VL (input) COMPLEX array, dimension (LDVL,M)
60 If JOB = 'E' or 'B', VL must contain left eigenvectors of T (or
61 of any Q*T*Q**H with Q unitary), corresponding to the eigen‐
62 pairs specified by HOWMNY and SELECT. The eigenvectors must be
63 stored in consecutive columns of VL, as returned by CHSEIN or
64 CTREVC. If JOB = 'V', VL is not referenced.
65 LDVL (input) INTEGER
67 The leading dimension of the array VL. LDVL >= 1; and if JOB =
68 'E' or 'B', LDVL >= N.
69 VR (input) COMPLEX array, dimension (LDVR,M)
71 If JOB = 'E' or 'B', VR must contain right eigenvectors of T
72 (or of any Q*T*Q**H with Q unitary), corresponding to the
73 eigenpairs specified by HOWMNY and SELECT. The eigenvectors
74 must be stored in consecutive columns of VR, as returned by
75 CHSEIN or CTREVC. If JOB = 'V', VR is not referenced.
76 LDVR (input) INTEGER
78 The leading dimension of the array VR. LDVR >= 1; and if JOB =
79 'E' or 'B', LDVR >= N.
80 S (output) REAL array, dimension (MM)
82 If JOB = 'E' or 'B', the reciprocal condition numbers of the
83 selected eigenvalues, stored in consecutive elements of the
84 array. Thus S(j), SEP(j), and the j-th columns of VL and VR all
85 correspond to the same eigenpair (but not in general the j-th
86 eigenpair, unless all eigenpairs are selected). If JOB = 'V',
87 S is not referenced.
88 SEP (output) REAL array, dimension (MM)
90 If JOB = 'V' or 'B', the estimated reciprocal condition numbers
91 of the selected eigenvectors, stored in consecutive elements of
92 the array. If JOB = 'E', SEP is not referenced.
93 MM (input) INTEGER
95 The number of elements in the arrays S (if JOB = 'E' or 'B')
96 and/or SEP (if JOB = 'V' or 'B'). MM >= M.
97 M (output) INTEGER
99 The number of elements of the arrays S and/or SEP actually used
100 to store the estimated condition numbers. If HOWMNY = 'A', M
101 is set to N.
102 WORK (workspace) COMPLEX array, dimension (LDWORK,N+6)
104 If JOB = 'E', WORK is not referenced.
105 LDWORK (input) INTEGER
107 The leading dimension of the array WORK. LDWORK >= 1; and if
108 JOB = 'V' or 'B', LDWORK >= N.
109 RWORK (workspace) REAL array, dimension (N)
111 If JOB = 'E', RWORK is not referenced.
112 INFO (output) INTEGER
114 = 0: successful exit
115 < 0: if INFO = -i, the i-th argument had an illegal value
116
118 The reciprocal of the condition number of an eigenvalue lambda is
119 defined as
120 S(lambda) = |v'*u| / (norm(u)*norm(v))
122 where u and v are the right and left eigenvectors of T corresponding to
124 lambda; v' denotes the conjugate transpose of v, and norm(u) denotes
125 the Euclidean norm. These reciprocal condition numbers always lie
126 between zero (very badly conditioned) and one (very well conditioned).
127 If n = 1, S(lambda) is defined to be 1.
128 An approximate error bound for a computed eigenvalue W(i) is given by
130 EPS * norm(T) / S(i)
132 where EPS is the machine precision.
134 The reciprocal of the condition number of the right eigenvector u cor‐
136 responding to lambda is defined as follows. Suppose
137 T = ( lambda c )
139 ( 0 T22 )
140 Then the reciprocal condition number is
142 SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )
144 where sigma-min denotes the smallest singular value. We approximate the
146 smallest singular value by the reciprocal of an estimate of the one-
147 norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is defined to
148 be abs(T(1,1)).
149 An approximate error bound for a computed right eigenvector VR(i) is
151 given by
152 EPS * norm(T) / SEP(i)
154 LAPACK routine (version 3.1) November 2006 CTRSNA(1)