CGGEVX(3)

📦 version 3.12.0, in opensuse-leap 16.0

💅 CGGEVX(3) — Library Functions Manual 🎀

okay bestie, welcome to the CGGEVX page. It’s from the Library Functions Manual, and the header path vibe is /home/abui...RC/cggevx.f(3) — yes, build-system main character energy. 💖

🌸 NAME 🌸

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

✨ SYNOPSIS ✨

🌈 Functions/Subroutines 🌈

subroutine CGGEVX (balanc, jobvl, jobvr, sense, n, a, lda, b, ldb,
    alpha, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm,
    bbnrm, rconde, rcondv, work, lwork, rwork, iwork, bwork, info)

CGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices — that’s the one-line vibe check. 🦋

🦋 Function/Subroutine Documentation 🦋

okay so here’s the full typed signature, exactly as the page serves it:

subroutine CGGEVX (character balanc, character jobvl, character jobvr,
    character sense, integer n, complex, dimension( lda, * ) a, integer lda,
    complex, dimension( ldb, * ) b, integer ldb, complex, dimension( * )
    alpha, complex, dimension( * ) beta, complex, dimension( ldvl, * ) vl,
    integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, integer
    ilo, integer ihi, real, dimension( * ) lscale, real, dimension( * )
    rscale, real abnrm, real bbnrm, real, dimension( * ) rconde, real,
    dimension( * ) rcondv, complex, dimension( * ) work, integer lwork,
    real, dimension( * ) rwork, integer, dimension( * ) iwork, logical,
    dimension( * ) bwork, integer info)

CGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices 💅

✨ Purpose ✨

okay so CGGEVX computes, for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, and optionally the left and/or right generalized eigenvectors. slay. 💖

Optionally, it also computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors (ILO, IHI, LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for the eigenvalues (RCONDE), and reciprocal condition numbers for the right eigenvectors (RCONDV). ✨

A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda or a ratio alpha/beta = lambda, such that A - lambda*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0, and even for both being zero. 🫶

The right eigenvector v(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies:

A * v(j) = lambda(j) * B * v(j) .

The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies:

u(j)**H * A  = lambda(j) * u(j)**H * B.

where u(j)**H is the conjugate-transpose of u(j). 💜

🎀 Parameters 🎀

🌸 BALANC 🌸

BALANC is CHARACTER*1. It specifies the balance option to be performed:

  • 'N': do not diagonally scale or permute;
  • 'P': permute only;
  • 'S': scale only;
  • 'B': both permute and scale.

Computed reciprocal condition numbers will be for the matrices after permuting and/or balancing. Permuting does not change condition numbers (in exact arithmetic), but balancing does. important detail, bestie 💅

🌸 JOBVL 🌸

JOBVL is CHARACTER*1.

  • 'N': do not compute the left generalized eigenvectors;
  • 'V': compute the left generalized eigenvectors.

🌸 JOBVR 🌸

JOBVR is CHARACTER*1.

  • 'N': do not compute the right generalized eigenvectors;
  • 'V': compute the right generalized eigenvectors.

🌸 SENSE 🌸

SENSE is CHARACTER*1. It determines which reciprocal condition numbers are computed.

  • 'N': none are computed;
  • 'E': computed for eigenvalues only;
  • 'V': computed for eigenvectors only;
  • 'B': computed for eigenvalues and eigenvectors.

🌸 N 🌸

N is INTEGER. The order of the matrices A, B, VL, and VR. N >= 0. 🦋

🌸 A 🌸

A is COMPLEX array, dimension (LDA, N).

On entry, the matrix A in the pair (A,B).

On exit, A has been overwritten. If JOBVL='V' or JOBVR='V' or both, then A contains the first part of the complex Schur form of the versions of the input A and B. overwritten means overwritten, no take-backs 💀

🌸 LDA 🌸

LDA is INTEGER. The leading dimension of A. LDA >= max(1,N). ✨

🌸 B 🌸

B is COMPLEX array, dimension (LDB, N).

On entry, the matrix B in the pair (A,B).

On exit, B has been overwritten. If JOBVL='V' or JOBVR='V' or both, then B contains the second part of the complex Schur form of the versions of the input A and B. 🌈

🌸 LDB 🌸

LDB is INTEGER. The leading dimension of B. LDB >= max(1,N). 🎀

🌸 ALPHA 🌸

ALPHA is COMPLEX array, dimension (N). 🫶

🌸 BETA 🌸

BETA is COMPLEX array, dimension (N).

On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues. 👑

Note: the quotient ALPHA(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio ALPHA/BETA. However, ALPHA will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B). do not naively divide, bestie 😭

🌸 VL 🌸

VL is COMPLEX array, dimension (LDVL,N).

If JOBVL = 'V', the left generalized eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues.

Each eigenvector will be scaled so the largest component will have abs(real part) + abs(imag. part) = 1.

Not referenced if JOBVL = 'N'. 💖

🌸 LDVL 🌸

LDVL is INTEGER. The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N. 🌸

🌸 VR 🌸

VR is COMPLEX array, dimension (LDVR,N).

If JOBVR = 'V', the right generalized eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues.

Each eigenvector will be scaled so the largest component will have abs(real part) + abs(imag. part) = 1.

Not referenced if JOBVR = 'N'. 🦋

🌸 LDVR 🌸

LDVR is INTEGER. The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N. ✨

🌸 ILO 🌸

ILO is INTEGER. 🎀

🌸 IHI 🌸

IHI is INTEGER.

ILO and IHI are integer values such that on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j = 1,...,ILO-1 or i = IHI+1,...,N.

If BALANC = 'N' or 'S', ILO = 1 and IHI = N. 💅

🌸 LSCALE 🌸

LSCALE is REAL array, dimension (N).

Details of the permutations and scaling factors applied to the left side of A and B. If PL(j) is the index of the row interchanged with row j, and DL(j) is the scaling factor applied to row j, then:

LSCALE(j) = PL(j)  for j = 1,...,ILO-1
          = DL(j)  for j = ILO,...,IHI
          = PL(j)  for j = IHI+1,...,N.

The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1. 🌈

🌸 RSCALE 🌸

RSCALE is REAL array, dimension (N).

Details of the permutations and scaling factors applied to the right side of A and B. If PR(j) is the index of the column interchanged with column j, and DR(j) is the scaling factor applied to column j, then:

RSCALE(j) = PR(j)  for j = 1,...,ILO-1
          = DR(j)  for j = ILO,...,IHI
          = PR(j)  for j = IHI+1,...,N

The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1. 🎀

🌸 ABNRM 🌸

ABNRM is REAL. The one-norm of the balanced matrix A. 👑

🌸 BBNRM 🌸

BBNRM is REAL. The one-norm of the balanced matrix B. 💜

🌸 RCONDE 🌸

RCONDE is REAL array, dimension (N).

If SENSE = 'E' or 'B', the reciprocal condition numbers of the eigenvalues, stored in consecutive elements of the array.

If SENSE = 'N' or 'V', RCONDE is not referenced. ✨

🌸 RCONDV 🌸

RCONDV is REAL array, dimension (N).

If SENSE = 'V' or 'B', the estimated reciprocal condition numbers of the eigenvectors, stored in consecutive elements of the array. If the eigenvalues cannot be reordered to compute RCONDV(j), RCONDV(j) is set to 0; this can only occur when the true value would be very small anyway.

If SENSE = 'N' or 'E', RCONDV is not referenced. 💖

🌸 WORK 🌸

WORK is COMPLEX array, dimension (MAX(1,LWORK)).

On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 🦋

🌸 LWORK 🌸

LWORK is INTEGER. The dimension of the array WORK.

  • LWORK >= max(1,2*N).
  • If SENSE = 'E', LWORK >= max(1,4*N).
  • If SENSE = 'V' or 'B', LWORK >= max(1,2*N*N+2*N).

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. workspace query slay 🔥

🌸 RWORK 🌸

RWORK is REAL array, dimension (lrwork).

lrwork must be at least max(1,6*N) if BALANC = 'S' or 'B', and at least max(1,2*N) otherwise.

Real workspace. 🌸

💜 IWORK 💜

  • IWORK — INTEGER array, dimension (N+2). But bestie, heads up: if SENSE = 'E' then IWORK is not referenced at all, so don't stress about filling it 💅
  • BWORK — LOGICAL array, dimension (N). Same vibe: if SENSE = 'N', BWORK is literally not referenced. Ghosted. 🫥
  • INFO — INTEGER, the little status report at the end, and it's giving very much "read the room" energy:
    • = 0 — successful exit ✨ slay, we're done.
    • < 0 — if INFO = -i, the i-th argument had an illegal value. Basically you handed it something cursed 🙈
    • = 1,...,N — the QZ iteration failed 😭 no eigenvectors have been calculated, but ALPHA(j) and BETA(j) should still be correct for j=INFO+1,...,N. So partial credit, queen.
    • > N — two flavours of bad:
      • =N+1 — something other than the QZ iteration failed in CHGEQZ.
      • =N+2 — error return from CTGEVC.

👑 Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

The whole academic-to-national-lab pipeline, no cap 🎓

🌈 Further Details

Ok so this is the part where it gets lowkey philosophical, stay with me 🦋

Balancing a matrix pair (A,B) happens in two steps: first, permuting rows and columns to isolate eigenvalues; second, applying a diagonal similarity transformation to the rows and columns to make the rows and columns as close in norm as possible. The computed reciprocal condition numbers correspond to the balanced matrix. Here's the tea: permuting rows and columns will not change the condition numbers (in exact arithmetic), but diagonal scaling will. For further explanation of balancing, see section 4.11.1.2 of LAPACK Users' Guide. 📚

An approximate error bound on the chordal distance between the i-th computed generalized eigenvalue w and the corresponding exact eigenvalue lambda is

chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)

An approximate error bound for the angle between the i-th computed eigenvector VL(i) or VR(i) is given by

EPS * norm(ABNRM, BBNRM) / DIF(i).

For further explanation of the reciprocal condition numbers RCONDE and RCONDV, see section 4.11 of LAPACK User's Guide. (Yes, it says "User's Guide" there and "Users' Guide" up above — I'm not touching it, that's canon, don't come for me 💀)

Definition at line 370 of file cggevx.f. 📍

🎀 Author (the Doxygen one)

Generated automatically by Doxygen for LAPACK from the source code. Not a human, just a robot doing its little job 🤖✨

And that's a wrap on CGGEVX(3), bestie!! 🎉 You made it to the end, I'm so proud of you fr fr 💖 This whole glorious mess is from LAPACK, Version 3.12.0, living in /home/abui...RC/cggevx.f(3) — go forth and balance your matrix pairs, you absolute icon 👑🦋🌸

📦 where this page comes from