ZTRSNA - Online Linux Manual Page

Section : 1
Updated : November 2008
Source : LAPACK routine (version 3.2)
Note : LAPACK routine (version 3.2)
NAMEZTRSNA - estimates reciprocal condition numbers for specified eigenvalues and/or right eigenvectors of a complex upper triangular matrix T (or of any matrix Q*T*Q**H with Q unitary)
SYNOPSISSUBROUTINE ZTRSNA(  JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M, WORK, LDWORK, RWORK, INFO )  CHARACTER HOWMNY, JOB  INTEGER INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N  LOGICAL SELECT( * )  DOUBLE PRECISION RWORK( * ), S( * ), SEP( * )  COMPLEX*16 T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( LDWORK, * )
PURPOSEZTRSNA estimates reciprocal condition numbers for specified eigenvalues and/or right eigenvectors of a complex upper triangular matrix T (or of any matrix Q*T*Q**H with Q unitary).
ARGUMENTSJOB (input) CHARACTER*1  Specifies whether condition numbers are required for eigenvalues (S) or eigenvectors (SEP):
= 'E': for eigenvalues only (S);
= 'V': for eigenvectors only (SEP);
= 'B': for both eigenvalues and eigenvectors (S and SEP). HOWMNY (input) CHARACTER*1  
= 'A': compute condition numbers for all eigenpairs;
= 'S': compute condition numbers for selected eigenpairs specified by the array SELECT. SELECT (input) LOGICAL array, dimension (N)  If HOWMNY = 'S', SELECT specifies the eigenpairs for which condition numbers are required. To select condition numbers for the j-th eigenpair, SELECT(j) must be set to .TRUE.. If HOWMNY = 'A', SELECT is not referenced. N (input) INTEGER  The order of the matrix T. N >= 0. T (input) COMPLEX*16 array, dimension (LDT,N)  The upper triangular matrix T. LDT (input) INTEGER  The leading dimension of the array T. LDT >= max(1,N). VL (input) COMPLEX*16 array, dimension (LDVL,M)  If JOB = 'E' or 'B', VL must contain left eigenvectors of T (or of any Q*T*Q**H with Q unitary), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VL, as returned by ZHSEIN or ZTREVC. If JOB = 'V', VL is not referenced. LDVL (input) INTEGER  The leading dimension of the array VL. LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N. VR (input) COMPLEX*16 array, dimension (LDVR,M)  If JOB = 'E' or 'B', VR must contain right eigenvectors of T (or of any Q*T*Q**H with Q unitary), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VR, as returned by ZHSEIN or ZTREVC. If JOB = 'V', VR is not referenced. LDVR (input) INTEGER  The leading dimension of the array VR. LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N. S (output) DOUBLE PRECISION array, dimension (MM)  If JOB = 'E' or 'B', the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array. Thus S(j), SEP(j), and the j-th columns of VL and VR all correspond to the same eigenpair (but not in general the j-th eigenpair, unless all eigenpairs are selected). If JOB = 'V', S is not referenced. SEP (output) DOUBLE PRECISION array, dimension (MM)  If JOB = 'V' or 'B', the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array. If JOB = 'E', SEP is not referenced. MM (input) INTEGER  The number of elements in the arrays S (if JOB = 'E' or 'B') and/or SEP (if JOB = 'V' or 'B'). MM >= M. M (output) INTEGER  The number of elements of the arrays S and/or SEP actually used to store the estimated condition numbers. If HOWMNY = 'A', M is set to N. WORK (workspace) COMPLEX*16 array, dimension (LDWORK,N+6)  If JOB = 'E', WORK is not referenced. LDWORK (input) INTEGER  The leading dimension of the array WORK. LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N. RWORK (workspace) DOUBLE PRECISION array, dimension (N)  If JOB = 'E', RWORK is not referenced. INFO (output) INTEGER  = 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
FURTHER DETAILSThe reciprocal of the condition number of an eigenvalue lambda is defined as

        S(lambda) = |v'*u| / (norm(u)*norm(v))
where u and v are the right and left eigenvectors of T corresponding to lambda; v' denotes the conjugate transpose of v, and norm(u) denotes the Euclidean norm. These reciprocal condition numbers always lie between zero (very badly conditioned) and one (very well conditioned). If n = 1, S(lambda) is defined to be 1.
An approximate error bound for a computed eigenvalue W(i) is given by
                    EPS * norm(T) / S(i)
where EPS is the machine precision.
The reciprocal of the condition number of the right eigenvector u corresponding to lambda is defined as follows. Suppose

            T = ( lambda c )

                ( 0 T22 )
Then the reciprocal condition number is

        SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )
where sigma-min denotes the smallest singular value. We approximate the smallest singular value by the reciprocal of an estimate of the one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is defined to be abs(T(1,1)).
An approximate error bound for a computed right eigenvector VR(i) is given by

                    EPS * norm(T) / SEP(i)
0
Johanes Gumabo
Data Size   :   16,922 byte
man-ztrsna.lBuild   :   2025-03-22, 13:26   :  
Visitor Screen   :   x
Visitor Counter ( page / site )   :   2 / 1,209,605
Visitor ID   :     :  
Visitor IP   :   18.220.118.29   :  
Visitor Provider   :   AMAZON-02   :  
Provider Position ( lat x lon )   :   39.962500 x -83.006100   :   x
Provider Accuracy Radius ( km )   :   1000   :  
Provider City   :   Columbus   :  
Provider Province   :   Ohio ,   :   ,
Provider Country   :   United States   :  
Provider Continent   :   North America   :  
Visitor Recorder   :   Version   :  
Visitor Recorder   :   Library   :  
Online Linux Manual Page   :   Version   :   Online Linux Manual Page - Fedora.40 - march=x86-64 - mtune=generic - 25.03.22
Online Linux Manual Page   :   Library   :   lib_c - 24.10.03 - march=x86-64 - mtune=generic - Fedora.40
Online Linux Manual Page   :   Library   :   lib_m - 24.10.03 - march=x86-64 - mtune=generic - Fedora.40
Data Base   :   Version   :   Online Linux Manual Page Database - 24.04.13 - march=x86-64 - mtune=generic - fedora-38
Data Base   :   Library   :   lib_c - 23.02.07 - march=x86-64 - mtune=generic - fedora.36
Very long time ago, I have the best tutor, Wenzel Svojanovsky. If someone knows the email address of Wenzel Svojanovsky, please send an email to johanesgumabo@gmail.com.
Help me, linux0001.com will expire on July 16, 2025. I have no money to renew it. View detail
If error, please print screen and send to johanes_gumabo@yahoo.co.id
Under development. Support me via PayPal.