TESTING/EIG/ddrgev3.f(3)

📦

version 3.12.1 · 3 versions

3.12.1 · arch rolling (latest)
3.12.0 · opensuse-leap 16.0
3.12.0 · centos-stream 10, fedora 43, fedora 44

💖 NAME 💖

okay so bestie, we're looking at TESTING/EIG/ddrgev3.f — and the manual page itself is TESTING/EIG/ddrgev3.f(3) 🫶. This is the tester page for DDRGEV3, and it's lowkey a whole journey. No cap. ✨

🌈 SYNOPSIS 🌈

Functions/Subroutines

subroutine ddrgev3 (nsizes, nn, ntypes, dotype, iseed, thresh, nounit,
     a, lda, b, s, t, q, ldq, z, qe, ldqe, alphar, alphai, beta, alphr1,
     alphi1, beta1, work, lwork, result, info)
DDRGEV3

💫 Function/Subroutine Documentation 💫

`subroutine ddrgev3`

This is the full typed signature, and yes it's long — it's giving "the whole friend group showed up" 🫶💅.

subroutine ddrgev3 (integer nsizes, integer, dimension( * ) nn, integer
     ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
     double precision thresh, integer nounit, double precision, dimension(
     lda, * ) a, integer lda, double precision, dimension( lda, * ) b, double
     precision, dimension( lda, * ) s, double precision, dimension( lda, * )
     t, double precision, dimension( ldq, * ) q, integer ldq, double
     precision, dimension( ldq, * ) z, double precision, dimension( ldqe, * )
     qe, integer ldqe, double precision, dimension( * ) alphar, double
     precision, dimension( * ) alphai, double precision, dimension( * ) beta,
     double precision, dimension( * ) alphr1, double precision, dimension( *
     ) alphi1, double precision, dimension( * ) beta1, double precision,
     dimension( * ) work, integer lwork, double precision, dimension( * )
     result, integer info)
DDRGEV3

🎯 Purpose 🎯

okay so DDRGEV3 checks the nonsymmetric generalized eigenvalue problem driver routine DGGEV3. It's the vibe-checker bestie. 💅

DGGEV3 computes, for a pair of n-by-n nonsymmetric matrices (A,B), the generalized eigenvalues and, optionally, the left and right eigenvectors.

A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A - w*B is singular. It is usually represented as the pair (alpha,beta), as there is reasonable interpretation for beta=0, and even for both being zero. (iconic little edge-case behavior, honestly) 👑

A right generalized eigenvector corresponding to a generalized eigenvalue w for a pair of matrices (A,B) is a vector r such that (A - wB) * r = 0. A left generalized eigenvector is a vector l such that l**H * (A - wB) = 0, where l**H is the conjugate-transpose of l.

When DDRGEV3 is called, a number of matrix () and a number of matrix are specified. For each size () and each type of matrix, a pair of matrices (A, B) will be generated and used for testing. For each matrix pair, the following tests will be performed and compared with the threshold THRESH.

Results from DGGEV3:

(1) max over all left eigenvalue/-vector pairs (alpha/beta,l) of

| VL**H * (beta A - alpha B) |/( ulp max(|beta A|, |alpha B|) )

where VL**H is the conjugate-transpose of VL.

(2) | |VL(i)| - 1 | / ulp and whether largest component real

VL(i) denotes the i-th column of VL.

(3) max over all left eigenvalue/-vector pairs (alpha/beta,r) of

| (beta A - alpha B) * VR | / ( ulp max(|beta A|, |alpha B|) )

(4) | |VR(i)| - 1 | / ulp and whether largest component real

VR(i) denotes the i-th column of VR.

(5) W(full) = W(partial)

W(full) denotes the eigenvalues computed when both l and r are also computed, and W(partial) denotes the eigenvalues computed when only W, only W and r, or only W and l are computed.

(6) VL(full) = VL(partial)

VL(full) denotes the left eigenvectors computed when both l and r are computed, and VL(partial) denotes the result when only l is computed.

(7) VR(full) = VR(partial)

VR(full) denotes the right eigenvectors computed when both l and r are also computed, and VR(partial) denotes the result when only l is computed.

🧪 Test Matrices 🧪

The sizes of the test matrices are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type will be generated. Currently, the list of possible types is:

(1)  ( 0, 0 )         (a pair of zero matrices)

(2)  ( I, 0 )         (an identity and a zero matrix)

(3)  ( 0, I )         (an identity and a zero matrix)

(4)  ( I, I )         (a pair of identity matrices)

        t   t
(5)  ( J , J  )       (a pair of transposed Jordan blocks)

                                    t                ( I   0  )
(6)  ( X, Y )         where  X = ( J   0  )  and Y = (      t )
                                 ( 0   I  )          ( 0   J  )
                      and I is a k x k identity and J a (k+1)x(k+1)
                      Jordan block; k=(N-1)/2

(7)  ( D, I )         where D is diag( 0, 1,..., N-1 ) (a diagonal
                      matrix with those diagonal entries.)
(8)  ( I, D )

(9)  ( big*D, small*I ) where  is near overflow and small=1/big

(10) ( small*D, big*I )

(11) ( big*I, small*D )

(12) ( small*I, big*D )

(13) ( big*D, big*I )

(14) ( small*D, small*I )

(15) ( D1, D2 )        where D1 is diag( 0, 0, 1, ..., N-3, 0 ) and
                       D2 is diag( 0, N-3, N-4,..., 1, 0, 0 )
          t   t
(16) Q ( J , J ) Z     where Q and Z are random orthogonal matrices.

(17) Q ( T1, T2 ) Z    where T1 and T2 are upper triangular matrices
                       with random O(1) entries above the diagonal
                       and diagonal entries diag(T1) =
                       ( 0, 0, 1, ..., N-3, 0 ) and diag(T2) =
                       ( 0, N-3, N-4,..., 1, 0, 0 )

(18) Q ( T1, T2 ) Z    diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 )
                       diag(T2) = ( 0, 1, 0, 1,..., 1, 0 )
                       s = machine precision.

(19) Q ( T1, T2 ) Z    diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 )
                       diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 )

                                                       N-5
(20) Q ( T1, T2 ) Z    diag(T1)=( 0, 0, 1, 1, a, ..., a   =s, 0 )
                       diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )

(21) Q ( T1, T2 ) Z    diag(T1)=( 0, 0, 1, r1, r2, ..., r(N-4), 0 )
                       diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
                       where r1,..., r(N-4) are random.

(22) Q ( big*T1, small*T2 ) Z    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(23) Q ( small*T1, big*T2 ) Z    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(24) Q ( small*T1, small*T2 ) Z  diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(25) Q ( big*T1, big*T2 ) Z      diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(26) Q ( T1, T2 ) Z     where T1 and T2 are random upper-triangular
                        matrices.

🎛️ Parameters 🎛️

`NSIZES`

NSIZES is INTEGER. It's the number of sizes of matrices to use. If it is zero, DDRGEV3 does nothing. Respect the boundary, bestie. NSIZES >= 0.

`NN`

NN is INTEGER array, dimension (NSIZES). An array containing the sizes to be used for the matrices. Zero values will be skipped. NN >= 0.

`NTYPES`

NTYPES is INTEGER. The number of elements in DOTYPE. If it is zero, DDRGEV3 does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .

`DOTYPE`

DOTYPE is LOGICAL array, dimension (NTYPES). If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored.

`ISEED`

ISEED is INTEGER array, dimension (4). On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to DDRGEV3 to continue the same random number sequence.

`THRESH`

THRESH is DOUBLE PRECISION. A test will count as if the , computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero.

`NOUNIT`

NOUNIT is INTEGER. The FORTRAN unit number for printing out error messages (e.g., if a routine returns IERR not equal to 0.)

`A`

Okay bestie, we're still in the argument-list era of this page — don't scroll away, these arrays are literally the whole plot 💅

A — A is DOUBLE PRECISION array, dimension(LDA, max(NN)). It's used to hold the original A matrix. It's used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. — so yes, that one very specific situation, iykyk ✨
LDA — LDA is INTEGER. It's the leading dimension of A, B, S, and T. It must be at least 1 and at least max( NN ). That's the rule, periodt 🎀
B — B is DOUBLE PRECISION array, dimension(LDA, max(NN)). It's used to hold the original B matrix. It's used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. — same exact vibe as A, no cap 🦋
S — S is DOUBLE PRECISION array, dimension (LDA, max(NN)). This is the Schur form matrix computed from A by DGGEV3. On exit, S contains the Schur form matrix corresponding to the matrix in A. She's giving computed-on-exit main character energy 👑
T — T is DOUBLE PRECISION array, dimension (LDA, max(NN)). This is the upper triangular matrix computed from B by DGGEV3. Upper triangular and unbothered 💅
Q — Q is DOUBLE PRECISION array, dimension (LDQ, max(NN)). This is the (left) eigenvectors matrix computed by DGGEV3. Left eigenvector representation, slay 🌸
LDQ — LDQ is INTEGER. It's the leading dimension of Q and Z. It must be at least 1 and at least max( NN ). Same energy as LDA, bestie 🫶
Z — Z is DOUBLE PRECISION array, dimension( LDQ, max(NN) ). This is the (right) orthogonal matrix computed by DGGEV3. Right-side representation, iconic ✨
QE — QE is DOUBLE PRECISION array, dimension( LDQ, max(NN) ). QE holds the computed right or left eigenvectors. It's literally holding the eigenvectors, okay 😭
LDQE — LDQE is INTEGER. It's the leading dimension of QE. LDQE >= max(1,max(NN)). Exact inequality, no improv 💖
ALPHAR — ALPHAR is DOUBLE PRECISION array, dimension (max(NN)). Real-part array, holding it down 🎀
ALPHAI — ALPHAI is DOUBLE PRECISION array, dimension (max(NN)). Imaginary-part array, also holding it down 🦋
BETA — BETA is DOUBLE PRECISION array, dimension (max(NN)). These are the generalized eigenvalues of (A,B) computed by DGGEV3. And the exact formula is: ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th generalized eigenvalue of A and B. Put that in your notes, bestie 📝
ALPHR1 — ALPHR1 is DOUBLE PRECISION array, dimension (max(NN)). Backup real-part array, lowkey essential 🌈
ALPHI1 — ALPHI1 is DOUBLE PRECISION array, dimension (max(NN)). Backup imaginary-part array, matching energy 🔥
BETA1 — BETA1 is DOUBLE PRECISION array, dimension (max(NN)). Like ALPHAR, ALPHAI, BETA, these arrays contain the eigenvalues of A and B, but those computed when DGGEV3 only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors. Partial decomposition crew, we respect it 💜
WORK — WORK is DOUBLE PRECISION array, dimension (LWORK). Workspace array, doing the behind-the-scenes work 🧚
LWORK — LWORK is INTEGER. It's the number of entries in WORK. And the requirement is exactly: LWORK >= MAX( 8*N, N*(N+1) ). Do not fumble the workspace math, fr fr ⭐
RESULT — RESULT is DOUBLE PRECISION array, dimension (2). These are the values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. So yes, capped at 1/ulp — overflow said not today 💀
INFO — INFO is INTEGER. The exit-status tea is:
- = 0: successful exit 🎉
- < 0: if INFO = -i, the i-th argument had an illegal value 🙈
- > 0: a routine returned an error code. INFO is the absolute value of the INFO value returned. So yes, absolute value, not vibes 💅

👑 Author

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

Definition at line 404 of file ddrgev3.f.

✨ Author

Generated automatically by Doxygen for LAPACK from the source code.

That's the whole page, bestie — LAPACK Version 3.12.0, TESTING/EIG/ddrgev3.f(3), and we're out 💅✨

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms