DDRGEV(3)

💖 DDRGEV(3) — Library Functions Manual 💖

okay bestie, this is the Library Functions Manual page for DDRGEV(3), and it is lowkey a whole journey. no cap, every detail is staying.

🌸 NAME 🌸

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/EIG/ddrgev.f

that's the name/path, and it's literally the whole NAME section. iconic.

✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

subroutine DDRGEV (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)
    DDRGEV

DDRGEV is the routine, periodt.

🦋 Function/Subroutine Documentation 🦋

okay so here's the full typed-out signature, exactly as the page gives it:

subroutine DDRGEV (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)
    DDRGEV

and the routine is DDRGEV, obviously. 💅

💖 Purpose 💖

DDRGEV checks the nonsymmetric generalized eigenvalue problem driver routine DGGEV.

DGGEV 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.

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 DDRGEV is called, a number of matrix sizes and a number of matrix types 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 DGGEV 🔥

here are the receipts from DGGEV, and yes you need all seven:

(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 types are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type j 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, DDRGES does nothing. Constraint: 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. Constraint: NN >= 0.

NTYPES

NTYPES is INTEGER. The number of elements in DOTYPE. If it is zero, DDRGES 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 DDRGES to continue the same random number sequence.

THRESH

THRESH is DOUBLE PRECISION. A test will count as a fail if the error, 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

A — this parameter's description starts here, and part 2 has the rest. 🫶

okay bestie, we're back in the parameter section and it's giving full cast-list energy 💅 Every argument below is part of the DDRGEV documentation, and yes, we're keeping every dimension, condition, and status code exactly as written — no cap.

A — A is DOUBLE PRECISION array, dimension(LDA, max(NN)). Used to hold the original A matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. 🫶
LDA — LDA is INTEGER. The leading dimension of A, B, S, and T. It must be at least 1 and at least max( NN ). ✨
B — B is DOUBLE PRECISION array, dimension(LDA, max(NN)). Used to hold the original B matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. 💖
S — S is DOUBLE PRECISION array, dimension (LDA, max(NN)). The Schur form matrix computed from A by DGGES. On exit, S contains the Schur form matrix corresponding to the matrix in A. iconic behavior, honestly 🌸
T — T is DOUBLE PRECISION array, dimension (LDA, max(NN)). The upper triangular matrix computed from B by DGGES. periodt 💅
Q — Q is DOUBLE PRECISION array, dimension (LDQ, max(NN)). The (left) eigenvectors matrix computed by DGGEV. left-eigenvector main character energy 👑
LDQ — LDQ is INTEGER. The leading dimension of Q and Z. It must be at least 1 and at least max( NN ). ✨
Z — Z is DOUBLE PRECISION array, dimension( LDQ, max(NN) ). The (right) orthogonal matrix computed by DGGES. right-orthogonal queen 🦋
QE — QE is DOUBLE PRECISION array, dimension( LDQ, max(NN) ). QE holds the computed right or left eigenvectors. no notes 🔥
LDQE — LDQE is INTEGER. The leading dimension of QE. LDQE >= max(1,max(NN)). 🌈
ALPHAR — ALPHAR is DOUBLE PRECISION array, dimension (max(NN)). That's the whole entry — just the array and its dimension, no extra lore. 🎀
ALPHAI — ALPHAI is DOUBLE PRECISION array, dimension (max(NN)). Same clean little dimension moment as above. 💜
BETA — BETA is DOUBLE PRECISION array, dimension (max(NN)). The generalized eigenvalues of (A,B) computed by DGGEV. ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th generalized eigenvalue of A and B. literally the eigenvalue formula, iykyk 🧚
ALPHR1 — ALPHR1 is DOUBLE PRECISION array, dimension (max(NN)). 🍓
ALPHI1 — ALPHI1 is DOUBLE PRECISION array, dimension (max(NN)). 🍓
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 DGGEV only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors. partial-eigendecomposition arc, fr fr 🌙
WORK — WORK is DOUBLE PRECISION array, dimension (LWORK). 🫶
LWORK — LWORK is INTEGER. The number of entries in WORK. LWORK >= MAX( 8*N, N*(N+1) ). math said what it said ✨
RESULT — RESULT is DOUBLE PRECISION array, dimension (2). The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. important lil safety moment 💖
INFO — INFO is INTEGER, and the status codes are:

= 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 if INFO is negative, it's an illegal-argument situation; if it's positive, a routine returned an error code. keep that in mind, bestie 💀

Author

This one's brought to you by the academic icons:

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

Definition at line 404 of file ddrgev.f. 🎀

Author (the Doxygen encore)

Generated automatically by Doxygen for LAPACK from the source code. slay, automation 💅

And that's the whole cast, bestie — every parameter, dimension, status code, author, and generated-by note is here, no cap 💖✨ Shoutout to LAPACK Version 3.12.0 and the source file /home/abui...IG/ddrgev.f (a.k.a. ddrgev.f) for the footer tea; no date on this one, but the documentation sparkle is timeless 🎀💅

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