trsna(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

💅 NAME

trsna — trsna: eig condition numbers (yes bestie, that's the whole name, it's giving minimalist icon behavior ✨)

🦋 SYNOPSIS

Functions

subroutine ctrsna (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr,
    s, sep, mm, m, work, ldwork, rwork, info)
    CTRSNA
subroutine dtrsna (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr,
    s, sep, mm, m, work, ldwork, iwork, info)
    DTRSNA
subroutine strsna (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr,
    s, sep, mm, m, work, ldwork, iwork, info)
    STRSNA
subroutine ztrsna (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr,
    s, sep, mm, m, work, ldwork, rwork, info)
    ZTRSNA

🌸 Detailed Description

Okay so this is the LAPACK girlie page for the *TRSNA family — four flavors, one per precision/type combo, and they all do the same job: reciprocal condition numbers for eigenvalues and/or right eigenvectors of a triangular (or quasi-triangular) matrix. Each one gets its own deep dive below. 👑

🎀 Function Documentation

subroutine ctrsna — the complex one, our main character 💜

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

CTRSNA

✨ Purpose

So like — CTRSNA 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 it, that's the whole vibe. 🫶

🔥 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. (She said "not referenced," periodt. 💀)
  • N — N is INTEGER. The order of the matrix T. N >= 0.
  • T — T is COMPLEX 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 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 CHSEIN or CTREVC. 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 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 CHSEIN or CTREVC. 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. 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. 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 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 REAL 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. (Yeah, defined to be 1, we don't argue with her 🌈)

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 ctrsna.f.

subroutine dtrsna — the double precision real one, she's giving classic 🎀

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

✨ Purpose

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).

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 cap, that's just the assignment. 🦋

🔥 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 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 —
🦋 and that's exactly where this part of the page cuts off — VL keeps going in part 2, bestie, don't go anywhere 💜
  • VL — okay so VL is a 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 and it’s the leading dimension of the array VL. LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.
  • VR — VR is a 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 and it’s the leading dimension of the array VR. LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.
  • S — S is a DOUBLE PRECISION array, dimension (MM). If JOB = 'E' or 'B', this is where the reciprocal condition numbers of the selected eigenvalues go, 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. So 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 a DOUBLE PRECISION array, dimension (MM). If JOB = 'V' or 'B', this holds 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 and it’s 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 and it’s 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 a DOUBLE PRECISION array, dimension (LDWORK,N+6). If JOB = 'E', WORK is not referenced.
  • LDWORK — LDWORK is INTEGER and it’s the leading dimension of the array WORK. LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.
  • IWORK — IWORK is an 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:

okay bestie, this is the condition-number lore: 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.

🌈 STRSNA

   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:

Bestie, 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).

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.

🎀 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 a 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. It’s the order of the matrix T. N >= 0.
  • T — T is a REAL array, dimension (LDT,N). It’s the upper quasi-triangular matrix T, in Schur canonical form.
  • LDT — LDT is INTEGER and it’s the leading dimension of the array T. LDT >= max(1,N).
  • VL — VL is a 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 and it’s the leading dimension of the array VL. LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.
  • VR — VR is a 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 and it’s the leading dimension of the array VR. LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.
  • S — S is a REAL array, dimension (MM). If JOB = 'E' or 'B', these are 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. So 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 a REAL array, dimension (MM). If JOB = 'V' or 'B', these are 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 — INTEGER, and it's the number of slots in the arrays S (only if JOB = 'E' or 'B') and/or SEP (only if JOB = 'V' or 'B'). Rule: MM >= M. That's it, that's the vibe 💅
  • M — INTEGER, the number of elements of S and/or SEP that actually got used to store the estimated condition numbers. And bestie, if HOWMNY = 'A', then M is set to N. Slay, it just does it for you ✨
  • WORK — REAL array, dimension (LDWORK,N+6). If JOB = 'E', WORK is not referenced. Not even a little. It just sits there 🙈
  • LDWORK — INTEGER, the leading dimension of the array WORK. LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N. Conditional queen 👑
  • IWORK — INTEGER array, dimension (2*(N-1)). If JOB = 'E', IWORK is not referenced. Same deal, it gets left on read 💀
  • INFO — INTEGER, the receipts:
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value. Yeah, that one's on you 🫶

💖 Author 💖

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🦋 Further Details 🦋

Okay so, 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). And if n = 1, S(lambda) is defined to be 1. Tiny matrix, perfect score, love that for it 🌸

An approximate error bound for a computed eigenvalue W(i) is given by

            EPS * norm(T) / S(i)

where EPS is the machine precision. iykyk ✨

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)). No cap, that's the whole definition 🧚

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. Noted, bookmarked, obsessed 📌

👑 ZTRSNA 👑

Here's the complex edition, and she's giving main character energy fr fr:

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). Same energy as the real one, just with complex numbers and a little more sparkle ✨

💜 Parameters 💜

  • JOB — 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). Both? Greedy. Respect 🫶
  • HOWMNY — CHARACTER*1:
    • = 'A': compute condition numbers for all eigenpairs;
    • = 'S': compute condition numbers for selected eigenpairs specified by the array SELECT.
  • SELECT — 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 — she's off duty 🙈
  • N — INTEGER, the order of the matrix T. N >= 0.
  • T — COMPLEX*16 array, dimension (LDT,N). The upper triangular matrix T. Upper triangular only, we don't do lower here 💅
  • LDT — INTEGER, the leading dimension of the array T. LDT >= max(1,N).
  • VL — 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 — INTEGER, the leading dimension of the array VL. LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.
  • VR — 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. Bestie she's literally just vibing when you don't need her 🌈
  • LDVR — INTEGER, the leading dimension of the array VR. LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.
  • S — 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). Read that again, it's important — the indices line up with the selected ones, not with the original numbering 👀
  • SEP — 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 — 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 — 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. She handles it, don't worry 💅
  • WORK — COMPLEX*16 array, dimension (LDWORK,N+6). If JOB = 'E', WORK is not referenced.
  • LDWORK — INTEGER, the leading dimension of the array WORK. LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.
  • RWORK — DOUBLE PRECISION array, dimension (N). If JOB = 'E', RWORK is not referenced. Yeah, she's the real-valued sidekick for the complex girlies 🍓
  • INFO — INTEGER, the final verdict:
    • = 0: successful exit 🎀
    • < 0: if INFO = -i, the i-th argument had an illegal value. Fix your argument bestie 😭

💖 Author 💖

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🦋 Further Details 🦋

So 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. Periodt 💅

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)). Same rules, conjugate-transpose edition 🧚‍♀️

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. Yes, a robot wrote this part, and honestly? She ate 🤖💖

Okay bestie that's the whole page — go condition some numbers, you've got this 🫶✨

That was LAPACK, Version 3.12.0, and this trsna(3) page is officially a wrap. Mwah! 💋🎀

📦 where this page comes from