ZTRSNA(3)

ZTRSNA(3) 💅 — Library Functions Manual ✨

NAME 📛

Okay bestie, this page is filed under the source path /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/ztrsna.f. That’s the name entry, no cap — it’s literally a path, and yes, it’s iconic in a deeply nerdy way.

SYNOPSIS 🦋

Functions/Subroutines

subroutine ZTRSNA (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr,
    s, sep, mm, m, work, ldwork, rwork, info)

ZTRSNA

Function/Subroutine Documentation 🧚

subroutine ZTRSNA (character job, character howmny, logical, dimension( * )
    select, integer n, complex*16, dimension( ldt, * ) t, integer ldt,
    complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16,
    dimension( ldvr, * ) vr, integer ldvr, double precision, dimension( * )
    s, double precision, dimension( * ) sep, integer mm, integer m,
    complex*16, dimension( ldwork, * ) work, integer ldwork, double
    precision, dimension( * ) rwork, integer info)

ZTRSNA

Purpose 🎯

ZTRSNA 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). That’s the whole assignment, and it ate.

Parameters 🔍

• JOB — JOB is 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 — HOWMNY is CHARACTER*1
  • = 'A': compute condition numbers for all eigenpairs;
  • = 'S': compute condition numbers for selected eigenpairs specified by the array SELECT.
• SELECT — SELECT is 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 — N is INTEGER. The order of the matrix T. N >= 0.
• T — T is COMPLEX*16 array, dimension (LDT,N). The upper triangular matrix T.
• LDT — LDT is INTEGER. The leading dimension of the array T. LDT >= max(1,N).
• VL — VL is 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 — LDVL is INTEGER. The leading dimension of the array VL. LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.
• VR — VR is 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 — LDVR is INTEGER. The leading dimension of the array VR. LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.
• S — S is 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 — SEP is 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 — MM is 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 — M is 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 — WORK is COMPLEX*16 array, dimension (LDWORK,N+6). If JOB = 'E', WORK is not referenced.
• LDWORK — LDWORK is INTEGER. The leading dimension of the array WORK. LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.
• RWORK — RWORK is DOUBLE PRECISION array, dimension (N). If JOB = 'E', RWORK is not referenced.
• INFO — INFO is INTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value

Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details 🧮

The reciprocal of the condition number of an eigenvalue lambda is defined as

S(lambda) = |v**H*u| / (norm(u)*norm(v))

where u and v are the right and left eigenvectors of T corresponding to lambda; v**H 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)

Definition at line 246 of file ztrsna.f.

Author (auto-generated 💅)

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Signed off with 💖 from the LAPACK source: LAPACK Version 3.12.0, file /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/ztrsna.f. That’s the footer vibe, bestie — periodt.