SRC/sggevx.f(3)

🦋 SRC/sggevx.f 🦋

from the Library Functions Manual — this is the SRC/sggevx.f(3) page, bestie, and it's a whole vibe.

🏷️ NAME

SRC/sggevx.f

✨ SYNOPSIS ✨

💖 Functions/Subroutines 💖

• subroutine sggevx (balanc, jobvl, jobvr, sense, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, work, lwork, iwork, bwork, info) — SGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices.

📜 Function/Subroutine Documentation 📜

subroutine sggevx (character balanc, character jobvl, character jobvr,
    character sense, integer n, real, dimension( lda, * ) a, integer lda,
    real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) alphar,
    real, dimension( * ) alphai, real, dimension( * ) beta, real, dimension(
    ldvl, * ) vl, integer ldvl, real, 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, real, dimension( * ) work, integer lwork,
    integer, dimension( * ) iwork, logical, dimension( * ) bwork, integer
    info)

SGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices.

🎯 Purpose 🎯

okay so SGGEVX computes, for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized eigenvalues — and optionally the left and/or right generalized eigenvectors. ✨

Optionally also, it 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.

• 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 REAL 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 real 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 REAL 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 real Schur form of the versions of the input A and B.
• LDB — LDB is INTEGER. The leading dimension of B. LDB >= max(1,N).
• ALPHAR — ALPHAR is REAL array, dimension (N).
• ALPHAI — ALPHAI is REAL array, dimension (N).
• BETA — BETA is REAL array, dimension (N). On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigenvalues. If ALPHAI(j) is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with ALPHAI(j+1) negative.

Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(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, ALPHAR and ALPHAI 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 REAL array, dimension (LDVL,N). If JOBVL = 'V', the left eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues. If the j-th eigenvalue is real, then u(j) = VL(:,j), the j-th column of VL. If the j-th and (j+1)-th eigenvalues form a complex conjugate pair, then u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1). Each eigenvector will be scaled so the largest component 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 REAL array, dimension (LDVR,N). If JOBVR = 'V', the right eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues. If the j-th eigenvalue is real, then v(j) = VR(:,j), the j-th column of VR. If the j-th and (j+1)-th eigenvalues form a complex conjugate pair, then v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1). Each eigenvector will be scaled so the largest component 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. For a complex conjugate pair of eigenvalues two consecutive elements of RCONDE are set to the same value. Thus RCONDE(j), RCONDV(j), and the j-th columns of VL and VR all correspond to the j-th eigenpair. If SENSE = 'N' or 'V', RCONDE is not referenced.
• RCONDV
• RCONDV — REAL array, dimension (N). If SENSE = 'V' or SENSE = 'B', this is where the estimated reciprocal condition numbers of the eigenvectors live, stored in consecutive elements of the array. For a complex eigenvector, two consecutive elements of RCONDV are set to the same value. If the eigenvalues cannot be reordered to compute RCONDV(j), then RCONDV(j) is set to 0; this can only occur when the true value would be very small anyway. If SENSE = 'N' or SENSE = 'E', RCONDV is not referenced. 💅

💼 WORK

WORK — REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. That's the little receipt LAPACK hands back. 🧾

🛠️ LWORK

LWORK — INTEGER. This is the dimension of the array WORK. The floor is LWORK >= max(1,2*N). Then, depending on the options:

• If BALANC = 'S' or BALANC = 'B', or JOBVL = 'V', or JOBVR = 'V', then LWORK >= max(1,6*N).
• If SENSE = 'E', then LWORK >= max(1,10*N).
• If SENSE = 'V' or SENSE = 'B', then LWORK >= 2*N*N+8*N+16.

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. ✨

🧮 IWORK

IWORK — INTEGER array, dimension (N+6). If SENSE = 'E', IWORK is not referenced.

🧚 BWORK

BWORK — LOGICAL array, dimension (N). If SENSE = 'N', BWORK is not referenced.

🚦 INFO

INFO — INTEGER. The vibe check:

• = 0: successful exit.
• < 0: if INFO = -i, the i-th argument had an illegal value.
• = 1,...,N: The QZ iteration failed. No eigenvectors have been calculated, but ALPHAR(j), ALPHAI(j), and BETA(j) should be correct for j=INFO+1,...,N.
• > N: =N+1: other than QZ iteration failed in SHGEQZ. =N+2: error return from STGEVC.

👑 Author

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

🔍 Further Details

Balancing a matrix pair (A,B) is a two-step glow-up: 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.

Definition at line 387 of file sggevx.f.

👑 Author

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💖 That's the whole tea, bestie — footer says LAPACK Version 3.12.0, source SRC/sggevx.f(3) (and ngl, it didn't give us a date, so no cap on that). 🎀