SRC/ctrsna.f(3)

Okay bestie, you’re on the SRC/ctrsna.f(3) page from the Library Functions Manual, and it’s highkey a whole linear-algebra moment 🌈💅

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🌸 NAME 🌸

SRC/ctrsna.f — that’s the name, periodt ✨

🌼 SYNOPSIS 🌼

💫 Functions/Subroutines 💫

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

💖 Function/Subroutine Documentation 💖

🌟 subroutine ctrsna 🌟

   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). That’s the whole assignment, no cap 💖

✨ Parameters ✨

• 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 — is CHARACTER*1:
  • 'A': compute condition numbers for all eigenpairs;
  • 'S': compute condition numbers for selected eigenpairs specified by the array 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.
• N — is INTEGER. The order of the matrix T. N >= 0.
• T — is COMPLEX array, dimension (LDT,N). The upper triangular matrix T.
• LDT — is INTEGER. The leading dimension of the array T. LDT >= max(1,N).
• 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 — is INTEGER. The leading dimension of the array VL. LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.
• 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 — is INTEGER. The leading dimension of the array VR. LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.
• 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 — 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 — 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 — 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 — is COMPLEX array, dimension (LDWORK,N+6). If JOB = 'E', WORK is not referenced.
• LDWORK — is INTEGER. The leading dimension of the array WORK. LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.
• RWORK — is REAL array, dimension (N). If JOB = 'E', RWORK is not referenced.
• 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.

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.

📜 Author 📜

Generated automatically by Doxygen for LAPACK from the source code.

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LAPACK Version 3.12.0 · SRC/ctrsna.f(3) — bye bestie, go estimate those condition numbers responsibly 💋🫶