STRSNA(3)

💅 STRSNA(3) 💅

okay bestie, you're in the LAPACK Library Functions Manual, and this page is STRSNA(3) — a whole subroutine about reciprocal condition numbers, no cap. Let's get into it 💖

🎀 NAME 🎀

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/strsna.f

✨ SYNOPSIS ✨

🦋 Functions/Subroutines 🦋

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

and the subroutine name is, shockingly, 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). Iconic behavior.

T must be in Schur canonical form (as returned by SHSEQR), that is, 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. That's the vibe check 💅

🎀 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 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).

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 🧚

Generated automatically by Doxygen for LAPACK from the source code.

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mwah 💖 this page is from LAPACK Version 3.12.0, source path /home/abui...RC/strsna.f(3); the footer had no separate date printed, so that's the whole cute sign-off, bestie 🫶