TESTING/EIG/cchkgg.f(3)

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💖 `TESTING/EIG/cchkgg.f(3)` — Library Functions Manual 💖

🌸 NAME 🌸

TESTING/EIG/cchkgg.f

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine cchkgg (nsizes, nn, ntypes, dotype, iseed, thresh, tstdif,
    thrshn, nounit, a, lda, b, h, t, s1, s2, p1, p2, u, ldu, v, q, z,
    alpha1, beta1, alpha3, beta3, evectl, evectr, work, lwork, rwork,
    llwork, result, info)

CCHKGG

🦋 Function/Subroutine Documentation 🦋

okay bestie, here is the full signature with every type and dimension laid out exactly:

subroutine cchkgg (integer nsizes, integer, dimension( * ) nn, integer
    ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
    real thresh, logical tstdif, real thrshn, integer nounit, complex,
    dimension( lda, * ) a, integer lda, complex, dimension( lda, * ) b,
    complex, dimension( lda, * ) h, complex, dimension( lda, * ) t, complex,
    dimension( lda, * ) s1, complex, dimension( lda, * ) s2, complex,
    dimension( lda, * ) p1, complex, dimension( lda, * ) p2, complex,
    dimension( ldu, * ) u, integer ldu, complex, dimension( ldu, * ) v,
    complex, dimension( ldu, * ) q, complex, dimension( ldu, * ) z, complex,
    dimension( * ) alpha1, complex, dimension( * ) beta1, complex,
    dimension( * ) alpha3, complex, dimension( * ) beta3, complex,
    dimension( ldu, * ) evectl, complex, dimension( ldu, * ) evectr,
    complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork,
    logical, dimension( * ) llwork, real, dimension( 15 ) result, integer
    info)

CCHKGG

🎀 Purpose 🎀

okay so CCHKGG is literally the vibe check for the nonsymmetric generalized eigenvalue problem routines. Like, it checks them. No cap.

CGGHRD factors A and B as U H V^H and U T V^H, where ^H means conjugate transpose, H is hessenberg, T is triangular, and U and V are unitary.

CHGEQZ factors H and T as Q S Z^H and Q P Z^H, where P and S are upper triangular and Q and Z are unitary. It also computes the generalized eigenvalues (alpha(1),beta(1)),...,(alpha(n),beta(n)), where alpha(j)=S(j,j) and beta(j)=P(j,j) -- thus, w(j) = alpha(j)/beta(j) is a root of the generalized eigenvalue problem

det( A - w(j) B ) = 0

and m(j) = beta(j)/alpha(j) is a root of the essentially equivalent problem

det( m(j) A - B ) = 0

CTGEVC computes the matrix L of left eigenvectors and the matrix R of right eigenvectors for the matrix pair ( S, P ). In the description below, l and r are left and right eigenvectors corresponding to the generalized eigenvalues (alpha,beta).

When CCHKGG is called, a number of matrix sizes (NSIZES) and a number of matrix types (NTYPES) are specified. For each size (NN(j)) and each type of matrix, one matrix will be generated and used to test the nonsymmetric eigenroutines. For each matrix, 13 tests will be performed. The first twelve should be small -- O(1). They will be compared with the threshold THRESH:

                  H
(1)   | A - U H V  | / ( |A| n ulp )

                  H
(2)   | B - U T V  | / ( |B| n ulp )

               H
(3)   | I - UU  | / ( n ulp )

               H
(4)   | I - VV  | / ( n ulp )

                  H
(5)   | H - Q S Z  | / ( |H| n ulp )

                  H
(6)   | T - Q P Z  | / ( |T| n ulp )

               H
(7)   | I - QQ  | / ( n ulp )

               H
(8)   | I - ZZ  | / ( n ulp )

(9)   max over all left eigenvalue/-vector pairs (beta/alpha,l) of
                           H
       | (beta A - alpha B) l | / ( ulp max( |beta A|, |alpha B| ) )

(10)  max over all left eigenvalue/-vector pairs (beta/alpha,l') of
                           H
       | (beta H - alpha T) l' | / ( ulp max( |beta H|, |alpha T| ) )

       where the eigenvectors l' are the result of passing Q to
       STGEVC and back transforming (JOB='B').

(11)  max over all right eigenvalue/-vector pairs (beta/alpha,r) of

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

(12)  max over all right eigenvalue/-vector pairs (beta/alpha,r') of

       | (beta H - alpha T) r' | / ( ulp max( |beta H|, |alpha T| ) )

       where the eigenvectors r' are the result of passing Z to
       STGEVC and back transforming (JOB='B').

The last three test ratios will usually be small, but there is no mathematical requirement that they be so. They are therefore compared with THRESH only if TSTDIF is .TRUE.

(13)  | S(Q,Z computed) - S(Q,Z not computed) | / ( |S| ulp )

(14)  | P(Q,Z computed) - P(Q,Z not computed) | / ( |P| ulp )

(15)  max( |alpha(Q,Z computed) - alpha(Q,Z not computed)|/|S| ,
           |beta(Q,Z computed) - beta(Q,Z not computed)|/|P| ) / ulp

In addition, the normalization of L and R are checked, and compared with the threshold THRSHN.

🧪 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. Literally nothing, iconic.
(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.
(5) ( J^t, J^t ) — a pair of transposed Jordan blocks.
(6) ( X, Y ) — where X = ( J^t 0 ; 0 I ) and Y = ( I 0 ; 0 J^t ), and I is a k x k identity and J a (k+1)x(k+1) Jordan block; k=(N-1)/2.

X = ( J^t   0 )     Y = ( I   0 )
    ( 0     I )         ( 0   J^t )

(7) ( D, I ) — where D is P*D1, P is a random unitary diagonal matrix (i.e., with random magnitude 1 entries on the diagonal), and D1=diag( 0, 1,..., N-1 ) (i.e., a diagonal matrix with D1(1,1)=0, D1(2,2)=1, ..., D1(N,N)=N-1.)
(8) ( I, D )
(9) ( big*D, small*I ) — where big 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=P*diag( 0, 0, 1, ..., N-3, 0 ) and D2=Q*diag( 0, N-3, N-4,..., 1, 0, 0 ), and P and Q are random unitary diagonal matrices.
(16) U ( J^t, J^t ) V — where U and V are random unitary matrices.
(17) U ( T1, T2 ) V — where T1 and T2 are upper triangular matrices with random O(1) entries above the diagonal and diagonal entries diag(T1) = P*( 0, 0, 1, ..., N-3, 0 ) and diag(T2) = Q*( 0, N-3, N-4,..., 1, 0, 0 )
(18) U ( T1, T2 ) V — diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 ), diag(T2) = ( 0, 1, 0, 1,..., 1, 0 ), s = machine precision.
(19) U ( T1, T2 ) V — diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 ), diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 )
(20) U ( T1, T2 ) V — diag(T1)=( 0, 0, 1, 1, a, ..., a^(N-5) =s, 0 ), diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
(21) U ( T1, T2 ) V — 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) U ( big*T1, small*T2 ) V — diag(T1) = P*( 0, 0, 1, ..., N-3, 0 ), diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(23) U ( small*T1, big*T2 ) V — diag(T1) = P*( 0, 0, 1, ..., N-3, 0 ), diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(24) U ( small*T1, small*T2 ) V — diag(T1) = P*( 0, 0, 1, ..., N-3, 0 ), diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(25) U ( big*T1, big*T2 ) V — diag(T1) = P*( 0, 0, 1, ..., N-3, 0 ), diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(26) U ( T1, T2 ) V — where T1 and T2 are random upper-triangular matrices.

🌈 Parameters 🌈

NSIZES

NSIZES is INTEGER

The number of sizes of matrices to use. If it is zero, CCHKGG does nothing. It must be at least zero. Like, zero is the whole shutdown button, bestie.

NN

NN is INTEGER array, dimension (NSIZES)

An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero.

NTYPES

NTYPES is INTEGER

The number of elements in DOTYPE. If it is zero, CCHKGG 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

okay so we're deep in the argument list now — every single parameter is getting its moment, bestie, no skips, no cap 💅✨

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. periodt.
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 CCHKGG to continue the same random number sequence. slay, deterministic chaos.
THRESH — THRESH is REAL. A test will count as a fail if the number, 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. keep it cute and small, bestie.
TSTDIF — TSTDIF is LOGICAL. Specifies whether test ratios 13-15 will be computed and compared with THRESH. = .FALSE.: Only test ratios 1-12 will be computed and tested. Ratios 13-15 will be set to zero. = .TRUE.: All the test ratios 1-15 will be computed and tested. choose your fighter.
THRSHN — THRSHN is REAL. Threshold for reporting eigenvector normalization error. If the normalization of any eigenvector differs from 1 by more than THRSHN*ulp, then a special error message will be printed. normalization vibe check 🫶

This is handled separately from the other tests, since only a compiler or programming error should cause an error message, at least if THRSHN is at least 5--10. like, only actual chaos should trigger this one.

NOUNIT — NOUNIT is INTEGER. The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.)
A — COMPLEX 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, H, T, S1, P1, S2, and P2. It must be at least 1 and at least max( NN ).
B — COMPLEX 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.
H — COMPLEX array, dimension (LDA, max(NN)). The upper Hessenberg matrix computed from A by CGGHRD. hessenberg era.
T — COMPLEX array, dimension (LDA, max(NN)). The upper triangular matrix computed from B by CGGHRD.
S1 — COMPLEX array, dimension (LDA, max(NN)). The Schur (upper triangular) matrix computed from H by CHGEQZ when Q and Z are also computed.
S2 — COMPLEX array, dimension (LDA, max(NN)). The Schur (upper triangular) matrix computed from H by CHGEQZ when Q and Z are not computed.
P1 — COMPLEX array, dimension (LDA, max(NN)). The upper triangular matrix computed from T by CHGEQZ when Q and Z are also computed.
P2 — COMPLEX array, dimension (LDA, max(NN)). The upper triangular matrix computed from T by CHGEQZ when Q and Z are not computed.
U — COMPLEX array, dimension (LDU, max(NN)). The (left) unitary matrix computed by CGGHRD.
LDU — LDU is INTEGER. The leading dimension of U, V, Q, Z, EVECTL, and EVECTR. It must be at least 1 and at least max( NN ).
V — COMPLEX array, dimension (LDU, max(NN)). The (right) unitary matrix computed by CGGHRD.
Q — COMPLEX array, dimension (LDU, max(NN)). The (left) unitary matrix computed by CHGEQZ.
Z — COMPLEX array, dimension (LDU, max(NN)). The (left) unitary matrix computed by CHGEQZ. yep, the source says left — we're not editing the lore.
ALPHA1 — COMPLEX array, dimension (max(NN)).
BETA1 — COMPLEX array, dimension (max(NN)). The generalized eigenvalues of (A,B) computed by CHGEQZ when Q, Z, and the full Schur matrices are computed.
ALPHA3 — COMPLEX array, dimension (max(NN)).
BETA3 — COMPLEX array, dimension (max(NN)). The generalized eigenvalues of (A,B) computed by CHGEQZ when neither Q, Z, nor the Schur matrices are computed.
EVECTL — COMPLEX array, dimension (LDU, max(NN)). The (lower triangular) left eigenvector matrix for the matrices in S1 and P1.
EVECTR — COMPLEX array, dimension (LDU, max(NN)). The (upper triangular) right eigenvector matrix for the matrices in S1 and P1.
WORK — COMPLEX array, dimension (LWORK).
LWORK — LWORK is INTEGER. The number of entries in WORK. This must be at least max( 4*N, 2 * N**2, 1 ), for all N=NN(j).
RWORK — REAL array, dimension (2*max(NN)).
LLWORK — LOGICAL array, dimension (max(NN)).
RESULT — REAL array, dimension (15). The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. protect the math, protect the vibes.
INFO — INFO is 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. exit status tea ☕

Author 👑

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

Definition at line 498 of file cchkgg.f.

Author (the generated one) 🤖

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