SRC/ztrevc3.f(3)

🌸 NAME 🌸

okay so bestie, this is the manual page for SRC/ztrevc3.f, straight out of the LAPACK Library Functions Manual. It's the home of the ztrevc3 subroutine, and she is literally doing the most for eigenvector girlies everywhere. 💅

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine ztrevc3 (side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr,
    mm, m, work, lwork, rwork, lrwork, info)
    ZTREVC3

🧚 Function/Subroutine Documentation 🧚

ztrevc3

subroutine ztrevc3 (character side, character howmny, logical, dimension( * ) select, integer n, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer lrwork, integer info)
    ZTREVC3

💖 Purpose 💖

okay so ZTREVC3 computes some or all of the right and/or left eigenvectors of a complex upper triangular matrix T. Matrices of this type are produced by the Schur factorization of a complex general matrix: A = Q*T*Q**H, as computed by ZHSEQR. no cap, that's the whole setup. ✨

The right eigenvector x and the left eigenvector y of T corresponding to an eigenvalue w are defined by:

T*x = w*x,     (y**H)*T = w*(y**H)

where y**H denotes the conjugate transpose of the vector y. The eigenvalues are not input to this routine, but are read directly from the diagonal of T. iconic behavior, honestly.

This routine returns the matrices X and/or Y of right and left eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an input matrix. If Q is the unitary factor that reduces a matrix A to Schur form T, then Q*X and Q*Y are the matrices of right and left eigenvectors of A. 🫶

This uses a Level 3 BLAS version of the back transformation. slay. 🔥

🎀 Parameters 🎀

• SIDE — is CHARACTER*1. = 'R': compute right eigenvectors only; = 'L': compute left eigenvectors only; = 'B': compute both right and left eigenvectors.
• HOWMNY — is CHARACTER*1. = 'A': compute all right and/or left eigenvectors; = 'B': compute all right and/or left eigenvectors, backtransformed using the matrices supplied in VR and/or VL; = 'S': compute selected right and/or left eigenvectors, as indicated by the logical array SELECT.
• SELECT — is LOGICAL array, dimension (N). If HOWMNY = 'S', SELECT specifies the eigenvectors to be computed. The eigenvector corresponding to the j-th eigenvalue is computed if SELECT(j) = .TRUE.. Not referenced if HOWMNY = 'A' or 'B'.
• N — is INTEGER. The order of the matrix T. N >= 0.
• T — is COMPLEX*16 array, dimension (LDT,N). The upper triangular matrix T. T is modified, but restored on exit.
• LDT — is INTEGER. The leading dimension of the array T. LDT >= max(1,N).
• VL — is COMPLEX*16 array, dimension (LDVL,MM). On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must contain an N-by-N matrix Q (usually the unitary matrix Q of Schur vectors returned by ZHSEQR). On exit, if SIDE = 'L' or 'B', VL contains: if HOWMNY = 'A', the matrix Y of left eigenvectors of T; if HOWMNY = 'B', the matrix Q*Y; if HOWMNY = 'S', the left eigenvectors of T specified by SELECT, stored consecutively in the columns of VL, in the same order as their eigenvalues. Not referenced if SIDE = 'R'.
• LDVL — is INTEGER. The leading dimension of the array VL. LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N.
• VR — is COMPLEX*16 array, dimension (LDVR,MM). On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must contain an N-by-N matrix Q (usually the unitary matrix Q of Schur vectors returned by ZHSEQR). On exit, if SIDE = 'R' or 'B', VR contains: if HOWMNY = 'A', the matrix X of right eigenvectors of T; if HOWMNY = 'B', the matrix Q*X; if HOWMNY = 'S', the right eigenvectors of T specified by SELECT, stored consecutively in the columns of VR, in the same order as their eigenvalues. Not referenced if SIDE = 'L'.
• LDVR — is INTEGER. The leading dimension of the array VR. LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N.
• MM — is INTEGER. The number of columns in the arrays VL and/or VR. MM >= M.
• M — is INTEGER. The number of columns in the arrays VL and/or VR actually used to store the eigenvectors. If HOWMNY = 'A' or 'B', M is set to N. Each selected eigenvector occupies one column.
• WORK — is COMPLEX*16 array, dimension (MAX(1,LWORK)).
• LWORK — is INTEGER. The dimension of array WORK. LWORK >= max(1,2*N). For optimum performance, LWORK >= N + 2*N*NB, where NB is the optimal blocksize. 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 — is DOUBLE PRECISION array, dimension (LRWORK).
• LRWORK — is INTEGER. The dimension of array RWORK. LRWORK >= max(1,N). If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the RWORK array, returns this value as the first entry of the RWORK array, and no error message related to LRWORK is issued by XERBLA.
• 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 algorithm used in this program is basically backward (forward) substitution, with scaling to make the code robust against possible overflow. very demure, very mindful. 🧚

Each eigenvector is normalized so that the element of largest magnitude has magnitude 1; here the magnitude of a complex number (x,y) is taken to be |x| + |y|. math ate that, fr fr. 💜

Definition at line 242 of file ztrevc3.f.

🫶 Author 🫶

Generated automatically by Doxygen for LAPACK from the source code.

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mwah 💋 that's the whole page, bestie. Sign-off vibes: LAPACK Version 3.12.0, SRC/ztrevc3.f(3). periodt. 🌈💖