TESTING/EIG/ddrgev.f(3)

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🌈 Bestie's Guide to `TESTING/EIG/ddrgev.f(3)` 🌈

okay so this is the Library Functions Manual page for TESTING/EIG/ddrgev.f(3), and we're about to make DDRGEV make sense. no cap, it's a whole situation, but we got this. 💅

✨ NAME ✨

TESTING/EIG/ddrgev.f — that's the file, bestie. the whole icon. 🫶

🌸 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

💖 Function/Subroutine Documentation 💖

   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

🦋 Purpose 🦋

DDRGEV checks the nonsymmetric generalized eigenvalue problem driver routine DGGEV. slay. 💅

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

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), because there is a reasonable interpretation for beta=0, and even for both being zero. iykyk. 🫶

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. literally the whole vibe. 🌸

When DDRGEV is called, a number of matrix () 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:

Test (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.

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

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

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

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

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

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

Test (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.

Test (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.

Test (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:

Type (1): ( 0, 0 ) — a pair of zero matrices. 💀
Type (2): ( I, 0 ) — an identity and a zero matrix.
Type (3): ( 0, I ) — an identity and a zero matrix.
Type (4): ( I, I ) — a pair of identity matrices.
Type (5): ( J^t, J^t ) — a pair of transposed Jordan blocks.
Type (6): ( X, Y ) — where the whole setup is:

                                     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

Type (7): ( D, I ) — where D is diag( 0, 1,..., N-1 ) (a diagonal matrix with those diagonal entries.)
Type (8): ( I, D )
Type (9): ( big*D, small*I ) — where is near overflow and small=1/big
Type (10): ( small*D, big*I )
Type (11): ( big*I, small*D )
Type (12): ( small*I, big*D )
Type (13): ( big*D, big*I )
Type (14): ( small*D, small*I )
Type (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 )
Type (16): Q ( J^t, J^t ) Z — where Q and Z are random orthogonal matrices.
Type (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 )
Type (18): Q ( T1, T2 ) Z — diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 ), diag(T2) = ( 0, 1, 0, 1,..., 1, 0 ), and s = machine precision.
Type (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 )
Type (20):

                                                        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 )

Type (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.
Type (22): Q ( big*T1, small*T2 ) Z — diag(T1) = ( 0, 0, 1, ..., N-3, 0 ), diag(T2) = ( 0, 1, ..., 1, 0, 0 )
Type (23): Q ( small*T1, big*T2 ) Z — diag(T1) = ( 0, 0, 1, ..., N-3, 0 ), diag(T2) = ( 0, 1, ..., 1, 0, 0 )
Type (24): Q ( small*T1, small*T2 ) Z — diag(T1) = ( 0, 0, 1, ..., N-3, 0 ), diag(T2) = ( 0, 1, ..., 1, 0, 0 )
Type (25): Q ( big*T1, big*T2 ) Z — diag(T1) = ( 0, 0, 1, ..., N-3, 0 ), diag(T2) = ( 0, 1, ..., 1, 0, 0 )
Type (26): Q ( T1, T2 ) Z — where T1 and T2 are random upper-triangular matrices.

💅 Parameters 💅

NSIZES — INTEGER. The number of sizes of matrices to use. If it is zero, DDRGES does nothing. NSIZES >= 0. okay so yes, it says DDRGES there. periodt. 🫶
NN — INTEGER array, dimension (NSIZES). An array containing the sizes to be used for the matrices. Zero values will be skipped. NN >= 0.
NTYPES — 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 — 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 — 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 — DOUBLE PRECISION. A test will count as a failure if the computed 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 — INTEGER. The FORTRAN unit number for printing out error messages (e.g., if a routine returns IERR not equal to 0.)
A — the part 1 page stops right after the parameter name A, so its full description is in the next part. 🫶

Okay so we’re still deep in the parameter list, and every single line here matters — no skimming, bestie. 💅

A — a DOUBLE PRECISION array, dimension(LDA, max(NN)). It holds the original A matrix. It’s used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. — those exact conditions, periodt.
LDA — INTEGER. The leading dimension of A, B, S, and T. It must be at least 1 and at least max( NN ). Respect the minimums, bestie. ✨
B — a DOUBLE PRECISION array, dimension(LDA, max(NN)). It holds the original B matrix. Just like A, it’s used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE.
S — a 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. Slay. 💖
T — a DOUBLE PRECISION array, dimension (LDA, max(NN)). The upper triangular matrix computed from B by DGGES. Iconic triangular behavior. 🎀
Q — a DOUBLE PRECISION array, dimension (LDQ, max(NN)). The (left) eigenvectors matrix computed by DGGEV. Left-eigenvector main character energy. 👑
LDQ — INTEGER. The leading dimension of Q and Z. It must be at least 1 and at least max( NN ). Do not play with it. 💀
Z — a DOUBLE PRECISION array, dimension( LDQ, max(NN) ). The (right) orthogonal matrix computed by DGGES. Right-orthogonal and ready. 🦋
QE — a DOUBLE PRECISION array, dimension( LDQ, max(NN) ). QE holds the computed right or left eigenvectors. It’s giving “eigenvectors went here.” 🌸
LDQE — INTEGER. The leading dimension of QE. LDQE >= max(1,max(NN)). That’s the rule, no cap. 🫶
ALPHAR — a DOUBLE PRECISION array, dimension (max(NN)). One-third of the generalized eigenvalue real/imag/beta trio. 💜
ALPHAI — a DOUBLE PRECISION array, dimension (max(NN)). The imaginary part bestie of the trio. 🌈
BETA — a 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. Math ate. 🔥
ALPHR1 — a DOUBLE PRECISION array, dimension (max(NN)). The backup-ish real-part array. ⭐
ALPHI1 — a DOUBLE PRECISION array, dimension (max(NN)). The backup-ish imaginary-part array. ✨
BETA1 — a 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. Important distinction, iykyk. 💅
WORK — a DOUBLE PRECISION array, dimension (LWORK). Workspace, obviously. 🧚
LWORK — INTEGER. The number of entries in WORK. LWORK >= MAX( 8*N, N*(N+1) ). Respect the workspace math, bestie — it’s not decorative. 😭
RESULT — a DOUBLE PRECISION array, dimension (2). The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. Safety first, overflow never. 🙈
INFO — INTEGER. = 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. Read the error code, don’t ignore it. 💀

👑 Author 👑

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

Definition at line 404 of file ddrgev.f.

👑 Author (yes, again, bestie) 👑

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Mwah 💋 — that’s the whole page, bestie: LAPACK Version 3.12.0, TESTING/EIG/ddrgev.f(3), signin’ off with maximum documentation energy. Periodt. 🌈✨

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