SRC/strsna.f(3)

📦
version 3.12.1 · 7 versions
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💅 SRC/strsna.f 💅

okay bestie, welcome to SRC/strsna.f, straight out of the Library Functions Manual — the page id is SRC/strsna.f(3), and we are about to make this Fortran subroutine make sense, fr fr. ✨

🌸 NAME 🌸

SRC/strsna.f — that’s the name, bestie. iconic. 💖

🦋 SYNOPSIS 🦋

✨ Functions/Subroutines ✨

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

🎀 Function/Subroutine Documentation 🎀

subroutine strsna (character job, character howmny, logical, dimension( * )
    select, integer n, real, dimension( ldt, * ) t, integer ldt, real,
    dimension( ldvl, * ) vl, integer ldvl, real, dimension( ldvr, * ) vr,
    integer ldvr, real, dimension( * ) s, real, dimension( * ) sep, integer
    mm, integer m, real, dimension( ldwork, * ) work, integer ldwork,
    integer, dimension( * ) iwork, integer info)
    STRSNA

🌈 Purpose 🌈

okay so STRSNA estimates reciprocal condition numbers for specified eigenvalues and/or right eigenvectors of a real upper quasi-triangular matrix T — or of any matrix Q*T*Q**T with Q orthogonal. no cap. 💅

T must be in Schur canonical form, as returned by SHSEQR. That means it’s block upper triangular with 1-by-1 and 2-by-2 diagonal blocks. Each 2-by-2 diagonal block has its diagonal elements equal and its off-diagonal elements of opposite sign. periodt. 🦋

💜 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 eigenpair corresponding to a real eigenvalue w(j), SELECT(j) must be set to .TRUE..

To select condition numbers corresponding to a complex conjugate pair of eigenvalues w(j) and w(j+1), either SELECT(j) or SELECT(j+1) or both 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 REAL array, dimension (LDT,N).

The upper quasi-triangular matrix T, in Schur canonical form.

🍓 LDT

LDT is INTEGER.

The leading dimension of the array T. LDT >= max(1,N).

🍓 VL

VL is REAL array, dimension (LDVL,M).

If JOB = 'E' or 'B', VL must contain left eigenvectors of T (or of any Q*T*Q**T with Q orthogonal), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VL, as returned by SHSEIN or STREVC.

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 REAL array, dimension (LDVR,M).

If JOB = 'E' or 'B', VR must contain right eigenvectors of T (or of any Q*T*Q**T with Q orthogonal), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VR, as returned by SHSEIN or STREVC.

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 REAL array, dimension (MM).

If JOB = 'E' or 'B', the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array. For a complex conjugate pair of eigenvalues two consecutive elements of S are set to the same value.

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. iykyk. 💀

If JOB = 'V', S is not referenced.

🍓 SEP

SEP is REAL array, dimension (MM).

If JOB = 'V' or 'B', the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array. For a complex eigenvector two consecutive elements of SEP are set to the same value.

If the eigenvalues cannot be reordered to compute SEP(j), SEP(j) is set to 0; this can only occur when the true value would be very small anyway.

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 REAL 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.

🍓 IWORK

IWORK is INTEGER array, dimension (2*(N-1)).

If JOB = 'E', IWORK 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**T*u| / (norm(u)*norm(v))

where u and v are the right and left eigenvectors of T corresponding to lambda; v**T denotes the 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 262 of file strsna.f. 👑

🧚 Author 🧚

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