SRC/cggevx.f(3)

📦

version 3.12.1 · 7 versions

3.12.1 · arch rolling (latest)
3.9.0 · centos-stream 9
3.10.0 · ubuntu jammy
3.12.0 · ubuntu noble
3.12.0 · ubuntu noble
3.12.0 · centos-stream 10, fedora 43, fedora 44
3.11.0 · debian bookworm

🌸 `SRC/cggevx.f(3)` — Library Functions Manual 🌸

okay bestie, welcome to the cggevx page — it’s giving complex generalized eigenvalues, eigenvectors, balancing lore, and condition numbers. iconic behavior, honestly. 💅

✨ NAME ✨

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

💖 Function/Subroutine Documentation 💖

   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

Bestie, 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. no cap.

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

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. periodt.

`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

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.

`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).

`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.

`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 is INTEGER array, dimension (N+2)

If SENSE = 'E', IWORK is not referenced.

`BWORK` 👀

BWORK — a LOGICAL array, dimension (N). If SENSE = 'N'? Yeah, BWORK is not referenced at all, she's off duty 💤
INFO — an INTEGER, and here's the whole gossip:
- = 0: successful exit, we love a clean win ✨
- < 0: if INFO = -i, the i-th argument had an illegal value (bestie, check your inputs 😭)
- = 1,...,N: the QZ iteration failed. No eigenvectors have been calculated, but ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N — so like, partial credit? 🙈
- > N:
  - =N+1: other than 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.

🧚 Further Details:

Okay this is the part where it gets lowkey deep, so stay with me 🫶

Balancing a matrix pair (A,B) includes, first, permuting rows and columns to isolate eigenvalues, second, applying 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. 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.

ngl those two formulas are giving "trust me bro the math works" energy, but they're literally the actual error bounds so write them down 📝

Definition at line 370 of file cggevx.f.

Okay bestie that's the whole page, we ate and left absolutely no crumbs 💅🦋 Generated automatically by Doxygen for LAPACK from the source code — LAPACK, Version 3.12.0, from SRC/cggevx.f(3). Go forth and diagonalize, icon 👑🌸

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
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SRC/cggevx.f(3)

🌸 SRC/cggevx.f(3) — Library Functions Manual 🌸

✨ NAME ✨

✨ SYNOPSIS ✨

Functions/Subroutines

💖 Function/Subroutine Documentation 💖

🎯 Purpose

📋 Parameters

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 👀