DTRSNA(3)

📦 version 3.12.0, in opensuse-leap 16.0

💅 DTRSNA(3) — Library Functions Manual 💅

okay bestie, welcome to the DTRSNA(3) page from the Library Functions Manual 💖 this is the whole glow-up, no cap.

🌸 NAME 🌸

This page is for DTRSNA(3), and the NAME line is literally this path, character for character:

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

✨ SYNOPSIS ✨

💅 Functions/Subroutines

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

DTRSNA — yes, that’s the main character energy right there 👑

🎀 Function/Subroutine Documentation 🎀

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

DTRSNA again, because the page said so and we respect the page ✨

🦋 Purpose

Okay so DTRSNA 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). That’s the whole vibe 💫

T must be in Schur canonical form (as returned by DHSEQR), 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. No shortcuts, bestie.

🌈 Parameters

  • JOB — okay so JOB is CHARACTER*1 and 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. Slay, no extra work 💅
  • N — N is INTEGER. The order of the matrix T. N >= 0.
  • T — T is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DHSEIN or DTREVC. 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 DOUBLE PRECISION 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 DHSEIN or DTREVC. 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. 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 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. 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 DOUBLE PRECISION 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 dtrsna.f.

👑 Author 👑

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