SRC/ztrsna.f(3)

🎀 NAME 🎀

SRC/ztrsna.f — okay bestie, this is the manual page for SRC/ztrsna.f(3) from the Library Functions Manual, and the main character is the ZTRSNA subroutine. Periodt. 💅

✨ SYNOPSIS ✨

Functions/Subroutines

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

• ZTRSNA — literally the icon, no cap. 🫶

🧚 Function/Subroutine Documentation 🧚

🌟 subroutine ztrsna

   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 🎯

Okay so 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). It's giving numerical self-care, fr fr. 💖

📝 Parameters 📝

JOB

• JOB is CHARACTER*1.
• It 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 👑

Generated automatically by Doxygen for LAPACK from the source code.

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That's the whole vibe, bestie 💖 This cutie is LAPACK Version 3.12.0, page SRC/ztrsna.f(3). Slay responsibly. 💅✨