CTRSNA(3)

✨ CTRSNA(3) — like, the Library Functions Manual one 💅

okay bestie, so the page you're looking at is the man page for /home/abui...RC/ctrsna.f(3), straight out of the Library Functions Manual, and honestly? It's giving linear algebra icon energy. Let's get into it 🫶

---

🌸 NAME 🌸

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

yep, that's the whole name, that's the file, we don't make the rules 💀

---

🎀 SYNOPSIS 🎀

Functions/Subroutines

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

CTRSNA ✨ (yes that's it, that's the whole description, she's mysterious like that)

---

🦋 Function/Subroutine Documentation 🦋

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 ⭐

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). 🔥

like, she's out here telling you how well-conditioned your eigenvalues are. no cap, that's the whole vibe check.

💜 Parameters 💜

• JOB — JOB is CHARACTER*1. Tells us whether we're doing condition numbers for eigenvalues (S) or eigenvectors (SEP), and omg you have options bestie:
  • = '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 j-th eigenpair, SELECT(j) must be set to .TRUE.. If HOWMNY = 'A', SELECT is not referenced — she's literally not invited to the party 🙈.
• N — N is INTEGER. The order of the matrix T. N >= 0. 🍓
• T — T is a COMPLEX array, dimension (LDT,N). The upper triangular matrix T. she's triangular, she's complex, she's the main character.
• LDT — LDT is INTEGER. The leading dimension of the array T. LDT >= max(1,N). ⭐
• VL — VL is a 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 a 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 a REAL array, dimension (MM). If JOB = 'E' or 'B', the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array. So like — 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). lowkey important, don't mix that up bestie 🫶. If JOB = 'V', S is not referenced.
• SEP — SEP is a 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. slay.
• WORK — WORK is a COMPLEX array, dimension (LDWORK,N+6). If JOB = 'E', WORK is not referenced. 💀 (six whole extra columns, she's expensive)
• LDWORK — LDWORK is INTEGER. The leading dimension of the array WORK. LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N. 🔥
• RWORK — RWORK is a 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.

the whole squad, respect 💜

🧚 Further Details: 🧚

The reciprocal of the condition number of an eigenvalue lambda is defined as fortran 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. 💅

okay so zero = messy, one = pristine. periodt.

An approximate error bound for a computed eigenvalue W(i) is given by fortran 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 fortran T = ( lambda c ) ( 0 T22 ) Then the reciprocal condition number is fortran 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 fortran EPS * norm(T) / SEP(i)

---

🎀 Definition at line 246 of file ctrsna.f. 🎀

🌟 Author 🌟

Generated automatically by Doxygen for LAPACK from the source code. (not a human, just a little robot doing her best 🤖💖)

---

and that's the whole page bestie!! mwah 💋

this absolute icon of a man page was brought to you by LAPACK, Version 3.12.0, dated /home/abui...RC/ctrsna.f(3) — go forth and condition your eigenvalues responsibly 🫶✨🎀