CGEEVX(3)

🌸 CGEEVX(3) — LAPACK Library Functions Manual 🌸

okay bestie, welcome to the CGEEVX(3) page from the LAPACK Library Functions Manual. This is the whole tea: eigenvalues, eigenvectors, balancing, condition numbers, and all the parameters. No cap, every detail is staying. 💅

✨ NAME ✨

The NAME section is literally just this path, no extra lore:

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

💫 SYNOPSIS 💫

Functions/Subroutines

Here’s the synopsis call, copied exactly so you can match it character-for-character:

subroutine CGEEVX (balanc, jobvl, jobvr, sense, n, a, lda, w, vl, ldvl,
    vr, ldvr, ilo, ihi, scale, abnrm, rconde, rcondv, work, lwork,
    rwork, info)

CGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices. That’s the one-liner, and it’s iconic. 🦋

🫶 Function/Subroutine Documentation 🫶

Full signature, bestie

subroutine CGEEVX (character balanc, character jobvl, character jobvr,
    character sense, integer n, complex, dimension( lda, * ) a, integer lda,
    complex, dimension( * ) w, complex, dimension( ldvl, * ) vl, integer
    ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, integer ilo,
    integer ihi, real, dimension( * ) scale, real abnrm, real, dimension( *
    ) rconde, real, dimension( * ) rcondv, complex, dimension( * ) work,
    integer lwork, real, dimension( * ) rwork, integer info)

CGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices. Same energy, now with the full parameter glow-up. ✨

🎯 Purpose 🎯

CGEEVX computes, for an N-by-N complex nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors.

Optionally also, it computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors (ILO, IHI, SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues (RCONDE), and reciprocal condition numbers for the right eigenvectors (RCONDV). It’s giving “eigen-everything.” 💖

The right eigenvector v(j) of A satisfies:

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

where lambda(j) is its eigenvalue.

The left eigenvector u(j) of A satisfies:

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

where u(j)**H denotes the conjugate transpose of u(j).

The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest component real. Slay normalization. 🎀

Balancing a matrix means permuting the rows and columns to make it more nearly upper triangular, and applying a diagonal similarity transformation D * A * D**(-1), where D is a diagonal matrix, to make its rows and columns closer in norm and the condition numbers of its eigenvalues and eigenvectors smaller. 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 4.10.2 of the LAPACK Users' Guide. iykyk — balancing is the main character here. 👑

🎀 Parameters 🎀

Okay, here’s the whole cast. Every parameter, every value, every default-vibe detail. 💅

• BALANC — BALANC is CHARACTER*1. Indicates how the input matrix should be diagonally scaled and/or permuted to improve the conditioning of its eigenvalues.
  • = 'N': Do not diagonally scale or permute;
  • = 'P': Perform permutations to make the matrix more nearly upper triangular. Do not diagonally scale;
  • = 'S': Diagonally scale the matrix, ie. replace A by D*A*D**(-1), where D is a diagonal matrix chosen to make the rows and columns of A more equal in norm. Do not permute;
  • = 'B': Both diagonally scale and permute A.
  • Computed reciprocal condition numbers will be for the matrix after balancing and/or permuting. Permuting does not change condition numbers (in exact arithmetic), but balancing does.
• JOBVL — JOBVL is CHARACTER*1.
  • = 'N': left eigenvectors of A are not computed;
  • = 'V': left eigenvectors of A are computed.
  • If SENSE = 'E' or 'B', JOBVL must = 'V'.
• JOBVR — JOBVR is CHARACTER*1.
  • = 'N': right eigenvectors of A are not computed;
  • = 'V': right eigenvectors of A are computed.
  • If SENSE = 'E' or 'B', JOBVR must = 'V'.
• SENSE — SENSE is CHARACTER*1. Determines which reciprocal condition numbers are computed.
  • = 'N': None are computed;
  • = 'E': Computed for eigenvalues only;
  • = 'V': Computed for right eigenvectors only;
  • = 'B': Computed for eigenvalues and right eigenvectors.
  • If SENSE = 'E' or 'B', both left and right eigenvectors must also be computed (JOBVL = 'V' and JOBVR = 'V').
• N — N is INTEGER. The order of the matrix A. N >= 0.
• A — A is COMPLEX array, dimension (LDA,N). On entry, the N-by-N matrix A. On exit, A has been overwritten. If JOBVL = 'V' or JOBVR = 'V', A contains the Schur form of the balanced version of the matrix A.
• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
• W — W is COMPLEX array, dimension (N). W contains the computed eigenvalues.
• VL — VL is COMPLEX 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 JOBVL = 'N', VL is not referenced. u(j) = VL(:,j), the j-th column of VL.
• LDVL — LDVL is INTEGER. The leading dimension of the array VL. LDVL >= 1; if JOBVL = 'V', LDVL >= N.
• VR — VR is COMPLEX 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 JOBVR = 'N', VR is not referenced. v(j) = VR(:,j), the j-th column of VR.
• LDVR — LDVR is INTEGER. The leading dimension of the array VR. LDVR >= 1; if JOBVR = 'V', LDVR >= N.
• ILO — ILO is INTEGER.
• IHI — IHI is INTEGER. ILO and IHI are integer values determined when A was balanced. The balanced A(i,j) = 0 if I > J and J = 1,...,ILO-1 or I = IHI+1,...,N.
• SCALE — SCALE is REAL array, dimension (N). Details of the permutations and scaling factors applied when balancing A. If P(j) is the index of the row and column interchanged with row and column j, and D(j) is the scaling factor applied to row and column j, then

SCALE(J) = P(J),    for J = 1,...,ILO-1
         = D(J),    for J = ILO,...,IHI
         = P(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 (the maximum of the sum of absolute values of elements of any column).
• RCONDE — RCONDE is REAL array, dimension (N). RCONDE(j) is the reciprocal condition number of the j-th eigenvalue.
• RCONDV — RCONDV is REAL array, dimension (N). RCONDV(j) is the reciprocal condition number of the j-th right eigenvector.
• 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. If SENSE = 'N' or 'E', LWORK >= max(1,2*N), and if SENSE = 'V' or 'B', LWORK >= N*N+2*N. For good performance, LWORK must generally be larger.
  • 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 (2*N).
• INFO — INFO is INTEGER.
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value.
  • > 0: if INFO = i, the QR algorithm failed to compute all the eigenvalues, and no eigenvectors or condition numbers have been computed; elements 1:ILO-1 and i+1:N of W contain eigenvalues which have converged.

🧑‍🎓 Author

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

Definition at line 285 of file cgeevx.f.

📜 Author

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