SRC/dgeev.f(3)

💅 SRC/dgeev.f 💅

okay so bestie, welcome to the glow-up of SRC/dgeev.f(3) — it lives in the Library Functions Manual and honestly? it's iconic. sit down, grab a strawberry drink 🍓, and let's get into it.

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✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

• subroutine dgeev (jobvl, jobvr, n, a, lda, wr, wi, vl, ldvl, vr, ldvr, work, lwork, info) — DGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

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🦋 Function/Subroutine Documentation 🦋

subroutine dgeev (character jobvl, character jobvr, integer n, double
    precision, dimension( lda, * ) a, integer lda, double precision,
    dimension( * ) wr, double precision, dimension( * ) wi, double
    precision, dimension( ldvl, * ) vl, integer ldvl, double precision,
    dimension( ldvr, * ) vr, integer ldvr, double precision, dimension( * )
    work, integer lwork, integer info)

DGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices — like, that's the whole vibe, she said it herself. 👑

🌸 Purpose 🌸

okay so here's the tea ☕:

DGEEV computes, for an N-by-N real nonsymmetric matrix A, the eigenvalues and (if you want them) the left and/or right eigenvectors.

The right eigenvector v(j) of A satisfies

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

where lambda(j) is its eigenvalue. cute, classic, no notes.

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). periodt. 💅

The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest component real. that's the flex, bestie — norm 1 and a real biggest component. ✨

🎀 Parameters 🎀

JOBVL

JOBVL is CHARACTER*1

• = 'N': left eigenvectors of A are not computed;
• = 'V': left eigenvectors of A are computed.

JOBVR

JOBVR is CHARACTER*1

• = 'N': right eigenvectors of A are not computed;
• = 'V': right eigenvectors of A are computed.

N

N is INTEGER

The order of the matrix A. N >= 0. (zero is allowed, she's chill like that.)

A

A is DOUBLE PRECISION array, dimension (LDA,N)

• On entry, the N-by-N matrix A.
• On exit, A has been overwritten. 😭 yes, she takes it, the original is gone, that's the deal.

LDA

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

WR

WR is DOUBLE PRECISION array, dimension (N)

(just vibes here, the real deets are down in WI 🫶)

WI

WI is DOUBLE PRECISION array, dimension (N)

WR and WI contain the real and imaginary parts, respectively, of the computed eigenvalues. Complex conjugate pairs of eigenvalues appear consecutively, with the eigenvalue having the positive imaginary part first. so they're sitting next to each other in the group chat, positive-imag friend first. 💖

VL

VL is DOUBLE PRECISION 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. (she's literally not invited, no hard feelings 🙈)
• 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)-st 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).

LDVL

LDVL is INTEGER

The leading dimension of the array VL. LDVL >= 1; if JOBVL = 'V', LDVL >= N.

VR

VR is DOUBLE PRECISION 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.
• 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)-st 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).

LDVR

LDVR is INTEGER

The leading dimension of the array VR. LDVR >= 1; if JOBVR = 'V', LDVR >= N.

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the optimal LWORK. she's telling you how much space she actually wanted. 🌈

LWORK

LWORK is INTEGER

The dimension of the array WORK. LWORK >= max(1,3*N), and if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N. For good performance, LWORK must generally be larger. (yes bestie, bigger. it's giving "give me room to breathe". 💀)

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. sneaky lil probe, she's just asking questions. 🧚

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 have been computed; elements i+1:N of WR and WI contain eigenvalues which have converged.

okay real talk ⚠️ — that > 0 case is not a slay. if it happens you get zero eigenvectors and only partial eigenvalues. read INFO before you trust anything downstream, fr fr.

👑 Author 👑

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

Definition at line 190 of file dgeev.f.

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💜 Author 💜

Generated automatically by Doxygen for LAPACK from the source code. (the bots did this one, no cap 🤖✨)

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and that's the whole page, bestie — LAPACK, Version 3.12.0, and yes this gorgeous thing is SRC/dgeev.f(3). go forth and diagonalize responsibly, ily 🫶💖🎀