TESTING/EIG/schkgg.f(3)

📦

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🎀 NAME 🎀

Bestie, this is TESTING/EIG/schkgg.f from the Library Functions Manual — section 3 energy. The function is schkgg, and yes, it is a whole mood.

✨ SYNOPSIS ✨

🧚 Functions/Subroutines

subroutine schkgg — the big signature, dropped raw:

subroutine schkgg (nsizes, nn, ntypes, dotype, iseed, thresh, tstdif,
    thrshn, nounit, a, lda, b, h, t, s1, s2, p1, p2, u, ldu, v, q, z,
    alphr1, alphi1, beta1, alphr3, alphi3, beta3, evectl, evectr, work,
    lwork, llwork, result, info)

SCHKGG — same routine, caps-lock main character energy.

💜 Function/Subroutine Documentation 💜

👑 subroutine `schkgg` / `SCHKGG` 👑

Typed signature, exactly:

subroutine schkgg (integer nsizes, integer, dimension( * ) nn, integer
    ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
    real thresh, logical tstdif, real thrshn, integer nounit, real,
    dimension( lda, * ) a, integer lda, real, dimension( lda, * ) b, real,
    dimension( lda, * ) h, real, dimension( lda, * ) t, real, dimension(
    lda, * ) s1, real, dimension( lda, * ) s2, real, dimension( lda, * ) p1,
    real, dimension( lda, * ) p2, real, dimension( ldu, * ) u, integer ldu,
    real, dimension( ldu, * ) v, real, dimension( ldu, * ) q, real,
    dimension( ldu, * ) z, real, dimension( * ) alphr1, real, dimension( * )
    alphi1, real, dimension( * ) beta1, real, dimension( * ) alphr3, real,
    dimension( * ) alphi3, real, dimension( * ) beta3, real, dimension( ldu,
    * ) evectl, real, dimension( ldu, * ) evectr, real, dimension( * ) work,
    integer lwork, logical, dimension( * ) llwork, real, dimension( 15 )
    result, integer info)

🦋 Purpose 🦋

Okay so SCHKGG is literally the vibe check for the nonsymmetric generalized eigenvalue problem routines. It checks SGGHRD, SHGEQZ, and STGEVC.

SGGHRD factors A and B as U H V^T and U T V^T, where the T superscript means transpose, H is hessenberg, T is triangular, and U and V are orthogonal.

SHGEQZ factors H and T as Q S Z^T and Q P Z^T, where P is upper triangular, S is in generalized Schur form (block upper triangular, with 1x1 and 2x2 blocks on the diagonal, the 2x2 blocks corresponding to complex conjugate pairs of generalized eigenvalues), and Q and Z are orthogonal. 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

STGEVC 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 SCHKGG 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, 15 tests will be performed. The first twelve should be small -- O(1). They will be compared with the threshold THRESH:

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

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

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

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

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

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

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

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

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

   | l**H * (beta S - alpha P) | / ( ulp max( |beta S|, |alpha P| ) )

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

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

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

      | (beta S - alpha T) r | / ( ulp max( |beta S|, |alpha T| ) )

(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 (HOWMNY='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
(5)  ( J , J  )       (a pair of transposed Jordan blocks)

(6) where X and Y are set up like this:

                                    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 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 is diag( 0, 0, 1, ..., N-3, 0 ) and D2 is diag( 0, N-3, N-4,..., 1, 0, 0 )
(16) where U and V are random orthogonal matrices:

           t   t
(16) U ( J , J ) V     where U and V are random orthogonal 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) = ( 0, 0, 1, ..., N-3, 0 ) and diag(T2) = ( 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 this exact diagonal setup:

                                                        N-5
(20) U ( T1, T2 ) V    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) = ( 0, 0, 1, ..., N-3, 0 ), diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(23) U ( small*T1, big*T2 ) V — diag(T1) = ( 0, 0, 1, ..., N-3, 0 ), diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(24) U ( small*T1, small*T2 ) V — diag(T1) = ( 0, 0, 1, ..., N-3, 0 ), diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(25) U ( big*T1, big*T2 ) V — diag(T1) = ( 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, SCHKGG does nothing. It must be at least zero.

`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, SCHKGG 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 — 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 SCHKGG to continue the same random number sequence. 🎲
THRESH — REAL. A test will count as [blank] if the [blank], computed as described above, exceeds THRESH (the source really does have blanks here, bestie — not me being chaotic). 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. 🚨
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 — 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. (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.) 🫶
NOUNIT — INTEGER. The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) 📠
A — REAL 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 — 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 — REAL 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 — REAL array, dimension (LDA, max(NN)). The upper Hessenberg matrix computed from A by SGGHRD. 🔥
T — REAL array, dimension (LDA, max(NN)). The upper triangular matrix computed from B by SGGHRD. 💜
S1 — REAL array, dimension (LDA, max(NN)). The Schur (block upper triangular) matrix computed from H by SHGEQZ when Q and Z are also computed. 🌈
S2 — REAL array, dimension (LDA, max(NN)). The Schur (block upper triangular) matrix computed from H by SHGEQZ when Q and Z are not computed. ⭐
P1 — REAL array, dimension (LDA, max(NN)). The upper triangular matrix computed from T by SHGEQZ when Q and Z are also computed. 🎀
P2 — REAL array, dimension (LDA, max(NN)). The upper triangular matrix computed from T by SHGEQZ when Q and Z are not computed. 🍓
U — REAL array, dimension (LDU, max(NN)). The (left) orthogonal matrix computed by SGGHRD. 👑
LDU — INTEGER. The leading dimension of U, V, Q, Z, EVECTL, and EVECTR. It must be at least 1 and at least max( NN ). 📐
V — REAL array, dimension (LDU, max(NN)). The (right) orthogonal matrix computed by SGGHRD. 💫
Q — REAL array, dimension (LDU, max(NN)). The (left) orthogonal matrix computed by SHGEQZ. 🧚
Z — REAL array, dimension (LDU, max(NN)). The (left) orthogonal matrix computed by SHGEQZ. 🪞
ALPHR1 — REAL array, dimension (max(NN)). 🌷
ALPHI1 — REAL array, dimension (max(NN)). 🌺
BETA1 — REAL array, dimension (max(NN)). The generalized eigenvalues of (A,B) computed by SHGEQZ when Q, Z, and the full Schur matrices are computed. On exit, ( ALPHR1(k)+ALPHI1(k)*i ) / BETA1(k) is the k-th generalized eigenvalue of the matrices in A and B. 🧮
ALPHR3 — REAL array, dimension (max(NN)). 🌼
ALPHI3 — REAL array, dimension (max(NN)). 🌻
BETA3 — REAL array, dimension (max(NN)). 🍯
EVECTL — REAL array, dimension (LDU, max(NN)). The (block lower triangular) left eigenvector matrix for the matrices in S1 and P1. (See STGEVC for the format.) 🫧
EVECTR — REAL array, dimension (LDU, max(NN)). The (block upper triangular) right eigenvector matrix for the matrices in S1 and P1. (See STGEVC for the format.) 💎
WORK — REAL array, dimension (LWORK). 🛠️
LWORK — INTEGER. The number of entries in WORK. This must be at least max( 2 * N**2, 6*N, 1 ), for all N=NN(j). 📦
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. 📊
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 506 of file schkgg.f. 📝

🧚 AUTHOR

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