ZCHKGG(3)

💖 ZCHKGG(3) — Library Functions Manual 💖

okay so bestie, this is ZCHKGG(3) from the Library Functions Manual — yes, the manual name is literally Library Functions Manual, and this page is giving serious LAPACK testing energy. ✨ The source path is right below in NAME, no cap. 💅

💖 NAME 💖

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

✨ SYNOPSIS ✨

🌸 Functions/Subroutines 🌸

       subroutine ZCHKGG (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)
           ZCHKGG

🫶 Function/Subroutine Documentation 🫶

💜 subroutine ZCHKGG 💜

   subroutine ZCHKGG (integer nsizes, integer, dimension( * ) nn, integer
       ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
       double precision thresh, logical tstdif, double precision thrshn,
       integer nounit, complex*16, dimension( lda, * ) a, integer lda,
       complex*16, dimension( lda, * ) b, complex*16, dimension( lda, * ) h,
       complex*16, dimension( lda, * ) t, complex*16, dimension( lda, * ) s1,
       complex*16, dimension( lda, * ) s2, complex*16, dimension( lda, * ) p1,
       complex*16, dimension( lda, * ) p2, complex*16, dimension( ldu, * ) u,
       integer ldu, complex*16, dimension( ldu, * ) v, complex*16, dimension(
       ldu, * ) q, complex*16, dimension( ldu, * ) z, complex*16, dimension( *
       ) alpha1, complex*16, dimension( * ) beta1, complex*16, dimension( * )
       alpha3, complex*16, dimension( * ) beta3, complex*16, dimension( ldu, *
       ) evectl, complex*16, dimension( ldu, * ) evectr, complex*16, dimension(
       * ) work, integer lwork, double precision, dimension( * ) rwork,
       logical, dimension( * ) llwork, double precision, dimension( 15 )
       result, integer info)
       ZCHKGG

🌸 Purpose 🌸

okay so ZCHKGG checks the nonsymmetric generalized eigenvalue problem routines. periodt. 💅 ZGGHRD factors A and B as U H V and U T V, where the little H up top means conjugate transpose — so H is hessenberg, T is triangular, and U and V are unitary. slay.

ZHGEQZ factors H and T as Q S Z and Q P Z, 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

ZTGEVC 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). iykyk. 🦋

When ZCHKGG is called, a number of matrix sizes () and a number of matrix types are specified. For each size () 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
       DTGEVC 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
       DTGEVC 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. 💀
(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) a pair of transposed Jordan blocks:

    t   t
( J , J  )

(6) ( X, Y ) — where

                                    t                ( I   0  )
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 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 , J ) V with transposed blocks:

      t   t
U ( J , J ) 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 — with

                                                      N-5
diag(T1)=( 0, 0, 1, 1, a, ..., a   =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. It's the number of sizes of matrices to use. If it is zero, ZCHKGG does nothing. It must be at least zero. ✨

NN

NN is INTEGER array, dimension (NSIZES). It's 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. It's the number of elements in DOTYPE. If it is zero, ZCHKGG 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 — okay so, this is a 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. the way this is literally the type-enable list 🎀
ISEED — seed central: 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 ZCHKGG to continue the same random number sequence. seed continuity slay 🌱
THRESH — threshold time: DOUBLE PRECISION. A test will count as a fail if the value 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. ⚠️ do not mess with zero
TSTDIF — 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.
THRSHN — precision police: DOUBLE PRECISION. 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. (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.) compiler oopsie detector 👀
NOUNIT — INTEGER. The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) error-message printer go brrr 🖨️
A — COMPLEX*16 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. conditional input moment 💅
LDA — 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 ). big enough or bust 📏
B — COMPLEX*16 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*16 array, dimension (LDA, max(NN)). The upper Hessenberg matrix computed from A by ZGGHRD. Hessenberg hive ✨
T — COMPLEX*16 array, dimension (LDA, max(NN)). The upper triangular matrix computed from B by ZGGHRD. 🌸
S1 — COMPLEX*16 array, dimension (LDA, max(NN)). The Schur (upper triangular) matrix computed from H by ZHGEQZ when Q and Z are also computed. 🦋
S2 — COMPLEX*16 array, dimension (LDA, max(NN)). The Schur (upper triangular) matrix computed from H by ZHGEQZ when Q and Z are not computed. 🎀
P1 — COMPLEX*16 array, dimension (LDA, max(NN)). The upper triangular matrix computed from T by ZHGEQZ when Q and Z are also computed. 💜
P2 — COMPLEX*16 array, dimension (LDA, max(NN)). The upper triangular matrix computed from T by ZHGEQZ when Q and Z are not computed. 🌈
U — COMPLEX*16 array, dimension (LDU, max(NN)). The (left) unitary matrix computed by ZGGHRD. 👑
LDU — INTEGER. The leading dimension of U, V, Q, Z, EVECTL, and EVEZTR. It must be at least 1 and at least max( NN ). yes, EVEZTR — source spelling, not me 💀
V — COMPLEX*16 array, dimension (LDU, max(NN)). The (right) unitary matrix computed by ZGGHRD. 🌀
Q — COMPLEX*16 array, dimension (LDU, max(NN)). The (left) unitary matrix computed by ZHGEQZ. 💫
Z — COMPLEX*16 array, dimension (LDU, max(NN)). The (left) unitary matrix computed by ZHGEQZ. both of them say left in the source, I’m just the messenger 💅
ALPHA1 — COMPLEX*16 array, dimension (max(NN)). 🍓
BETA1 — COMPLEX*16 array, dimension (max(NN)). The generalized eigenvalues of (A,B) computed by ZHGEQZ when Q, Z, and the full Schur matrices are computed. 🔥
ALPHA3 — COMPLEX*16 array, dimension (max(NN)). 🧚
BETA3 — COMPLEX*16 array, dimension (max(NN)). The generalized eigenvalues of (A,B) computed by ZHGEQZ when neither Q, Z, nor the Schur matrices are computed. ⭐
EVECTL — COMPLEX*16 array, dimension (LDU, max(NN)). The (lower triangular) left eigenvector matrix for the matrices in S1 and P1. 💖
EVECTR — COMPLEX*16 array, dimension (LDU, max(NN)). The (upper triangular) right eigenvector matrix for the matrices in S1 and P1. 🫶
WORK — COMPLEX*16 array, dimension (LWORK). 🛠️
LWORK — INTEGER. The number of entries in WORK. This must be at least max( 4*N, 2 * N**2, 1 ), for all N=NN(j). work work work work work 💅
RWORK — DOUBLE PRECISION array, dimension (2*max(NN)). 🌸
LLWORK — LOGICAL array, dimension (max(NN)). 🎀
RESULT — DOUBLE PRECISION array, dimension (15). The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. capped for safety 🔥
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.

👑 Author

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

Definition at line 498 of file zchkgg.f. 🫶

👑 Author

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