---
name: "zchkhs.f"
section: "3"
platform: "linux"
package: "lapack-doc"
version: "3.12.1-2"
release: "Arch Linux 2026-11-01"
platforms: ["linux"]
url: "https://new.carta.tech/zchkhs.f"
---

# zchkhs.f(3)

## NAME

TESTING/EIG/zchkhs.f

## SYNOPSIS

### Functions/Subroutines

subroutine **zchkhs** (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a, lda, h, t1, t2, u, ldu, z, uz, w1, w3, evectl, evectr, evecty, evectx, uu, tau, work, nwork, rwork, iwork, select, result, info)\
 **ZCHKHS**

## Function/Subroutine Documentation

### subroutine zchkhs (integer nsizes, integer, dimension( \* ) nn, integer ntypes, logical, dimension( \* ) dotype, integer, dimension( 4 ) iseed, double precision thresh, integer nounit, complex\*16, dimension( lda, \* ) a, integer lda, complex\*16, dimension( lda, \* ) h, complex\*16, dimension( lda, \* ) t1, complex\*16, dimension( lda, \* ) t2, complex\*16, dimension( ldu, \* ) u, integer ldu, complex\*16, dimension( ldu, \* ) z, complex\*16, dimension( ldu, \* ) uz, complex\*16, dimension( \* ) w1, complex\*16, dimension( \* ) w3, complex\*16, dimension( ldu, \* ) evectl, complex\*16, dimension( ldu, \* ) evectr, complex\*16, dimension( ldu, \* ) evecty, complex\*16, dimension( ldu, \* ) evectx, complex\*16, dimension( ldu, \* ) uu, complex\*16, dimension( \* ) tau, complex\*16, dimension( \* ) work, integer nwork, double precision, dimension( \* ) rwork, integer, dimension( \* ) iwork, logical, dimension( \* ) select, double precision, dimension( 16 ) result, integer info)

**ZCHKHS**

**Purpose:**

```
!>
!>    ZCHKHS  checks the nonsymmetric eigenvalue problem routines.
!>
!>            ZGEHRD factors A as  U H U' , where ' means conjugate
!>            transpose, H is hessenberg, and U is unitary.
!>
!>            ZUNGHR generates the unitary matrix U.
!>
!>            ZUNMHR multiplies a matrix by the unitary matrix U.
!>
!>            ZHSEQR factors H as  Z T Z' , where Z is unitary and T
!>            is upper triangular.  It also computes the eigenvalues,
!>            w(1), ..., w(n); we define a diagonal matrix W whose
!>            (diagonal) entries are the eigenvalues.
!>
!>            ZTREVC computes the left eigenvector matrix L and the
!>            right eigenvector matrix R for the matrix T.  The
!>            columns of L are the complex conjugates of the left
!>            eigenvectors of T.  The columns of R are the right
!>            eigenvectors of T.  L is lower triangular, and R is
!>            upper triangular.
!>
!>            ZHSEIN computes the left eigenvector matrix Y and the
!>            right eigenvector matrix X for the matrix H.  The
!>            columns of Y are the complex conjugates of the left
!>            eigenvectors of H.  The columns of X are the right
!>            eigenvectors of H.  Y is lower triangular, and X is
!>            upper triangular.
!>
!>            ZTREVC3 computes left and right eigenvector matrices
!>            from a Schur matrix T and backtransforms them with Z
!>            to eigenvector matrices L and R for A. L and R are
!>            GE matrices.
!>
!>    When ZCHKHS is called, a number of matrix  () and a
!>    number of matrix  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, 16
!>    tests will be performed:
!>
!>    (1)     | A - U H U**H | / ( |A| n ulp )
!>
!>    (2)     | I - UU**H | / ( n ulp )
!>
!>    (3)     | H - Z T Z**H | / ( |H| n ulp )
!>
!>    (4)     | I - ZZ**H | / ( n ulp )
!>
!>    (5)     | A - UZ H (UZ)**H | / ( |A| n ulp )
!>
!>    (6)     | I - UZ (UZ)**H | / ( n ulp )
!>
!>    (7)     | T(Z computed) - T(Z not computed) | / ( |T| ulp )
!>
!>    (8)     | W(Z computed) - W(Z not computed) | / ( |W| ulp )
!>
!>    (9)     | TR - RW | / ( |T| |R| ulp )
!>
!>    (10)    | L**H T - W**H L | / ( |T| |L| ulp )
!>
!>    (11)    | HX - XW | / ( |H| |X| ulp )
!>
!>    (12)    | Y**H H - W**H Y | / ( |H| |Y| ulp )
!>
!>    (13)    | AX - XW | / ( |A| |X| ulp )
!>
!>    (14)    | Y**H A - W**H Y | / ( |A| |Y| ulp )
!>
!>    (15)    | AR - RW | / ( |A| |R| ulp )
!>
!>    (16)    | LA - WL | / ( |A| |L| ulp )
!>
!>    The  are specified by an array NN(1:NSIZES); the value of
!>    each element NN(j) specifies one size.
!>    The  are specified by a logical array DOTYPE( 1:NTYPES );
!>    if DOTYPE(j) is .TRUE., then matrix type  will be generated.
!>    Currently, the list of possible types is:
!>
!>    (1)  The zero matrix.
!>    (2)  The identity matrix.
!>    (3)  A (transposed) Jordan block, with 1's on the diagonal.
!>
!>    (4)  A diagonal matrix with evenly spaced entries
!>         1, ..., ULP  and random complex angles.
!>         (ULP = (first number larger than 1) - 1 )
!>    (5)  A diagonal matrix with geometrically spaced entries
!>         1, ..., ULP  and random complex angles.
!>    (6)  A diagonal matrix with  entries 1, ULP, ..., ULP
!>         and random complex angles.
!>
!>    (7)  Same as (4), but multiplied by SQRT( overflow threshold )
!>    (8)  Same as (4), but multiplied by SQRT( underflow threshold )
!>
!>    (9)  A matrix of the form  U' T U, where U is unitary and
!>         T has evenly spaced entries 1, ..., ULP with random complex
!>         angles on the diagonal and random O(1) entries in the upper
!>         triangle.
!>
!>    (10) A matrix of the form  U' T U, where U is unitary and
!>         T has geometrically spaced entries 1, ..., ULP with random
!>         complex angles on the diagonal and random O(1) entries in
!>         the upper triangle.
!>
!>    (11) A matrix of the form  U' T U, where U is unitary and
!>         T has  entries 1, ULP,..., ULP with random
!>         complex angles on the diagonal and random O(1) entries in
!>         the upper triangle.
!>
!>    (12) A matrix of the form  U' T U, where U is unitary and
!>         T has complex eigenvalues randomly chosen from
!>         ULP < |z| < 1   and random O(1) entries in the upper
!>         triangle.
!>
!>    (13) A matrix of the form  X' T X, where X has condition
!>         SQRT( ULP ) and T has evenly spaced entries 1, ..., ULP
!>         with random complex angles on the diagonal and random O(1)
!>         entries in the upper triangle.
!>
!>    (14) A matrix of the form  X' T X, where X has condition
!>         SQRT( ULP ) and T has geometrically spaced entries
!>         1, ..., ULP with random complex angles on the diagonal
!>         and random O(1) entries in the upper triangle.
!>
!>    (15) A matrix of the form  X' T X, where X has condition
!>         SQRT( ULP ) and T has  entries 1, ULP,..., ULP
!>         with random complex angles on the diagonal and random O(1)
!>         entries in the upper triangle.
!>
!>    (16) A matrix of the form  X' T X, where X has condition
!>         SQRT( ULP ) and T has complex eigenvalues randomly chosen
!>         from   ULP < |z| < 1   and random O(1) entries in the upper
!>         triangle.
!>
!>    (17) Same as (16), but multiplied by SQRT( overflow threshold )
!>    (18) Same as (16), but multiplied by SQRT( underflow threshold )
!>
!>    (19) Nonsymmetric matrix with random entries chosen from |z| < 1
!>    (20) Same as (19), but multiplied by SQRT( overflow threshold )
!>    (21) Same as (19), but multiplied by SQRT( underflow threshold )
!>
```

```
!>  NSIZES - INTEGER
!>           The number of sizes of matrices to use.  If it is zero,
!>           ZCHKHS does nothing.  It must be at least zero.
!>           Not modified.
!>
!>  NN     - 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.
!>           Not modified.
!>
!>  NTYPES - INTEGER
!>           The number of elements in DOTYPE.   If it is zero, ZCHKHS
!>           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. .
!>           Not modified.
!>
!>  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.
!>           Not modified.
!>
!>  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 ZCHKHS to continue the same random number
!>           sequence.
!>           Modified.
!>
!>  THRESH - DOUBLE PRECISION
!>           A test will count as  if the , 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.
!>           Not modified.
!>
!>  NOUNIT - INTEGER
!>           The FORTRAN unit number for printing out error messages
!>           (e.g., if a routine returns IINFO not equal to 0.)
!>           Not modified.
!>
!>  A      - COMPLEX*16 array, dimension (LDA,max(NN))
!>           Used to hold the matrix whose eigenvalues are to be
!>           computed.  On exit, A contains the last matrix actually
!>           used.
!>           Modified.
!>
!>  LDA    - INTEGER
!>           The leading dimension of A, H, T1 and T2.  It must be at
!>           least 1 and at least max( NN ).
!>           Not modified.
!>
!>  H      - COMPLEX*16 array, dimension (LDA,max(NN))
!>           The upper hessenberg matrix computed by ZGEHRD.  On exit,
!>           H contains the Hessenberg form of the matrix in A.
!>           Modified.
!>
!>  T1     - COMPLEX*16 array, dimension (LDA,max(NN))
!>           The Schur (=) matrix computed by ZHSEQR
!>           if Z is computed.  On exit, T1 contains the Schur form of
!>           the matrix in A.
!>           Modified.
!>
!>  T2     - COMPLEX*16 array, dimension (LDA,max(NN))
!>           The Schur matrix computed by ZHSEQR when Z is not computed.
!>           This should be identical to T1.
!>           Modified.
!>
!>  LDU    - INTEGER
!>           The leading dimension of U, Z, UZ and UU.  It must be at
!>           least 1 and at least max( NN ).
!>           Not modified.
!>
!>  U      - COMPLEX*16 array, dimension (LDU,max(NN))
!>           The unitary matrix computed by ZGEHRD.
!>           Modified.
!>
!>  Z      - COMPLEX*16 array, dimension (LDU,max(NN))
!>           The unitary matrix computed by ZHSEQR.
!>           Modified.
!>
!>  UZ     - COMPLEX*16 array, dimension (LDU,max(NN))
!>           The product of U times Z.
!>           Modified.
!>
!>  W1     - COMPLEX*16 array, dimension (max(NN))
!>           The eigenvalues of A, as computed by a full Schur
!>           decomposition H = Z T Z'.  On exit, W1 contains the
!>           eigenvalues of the matrix in A.
!>           Modified.
!>
!>  W3     - COMPLEX*16 array, dimension (max(NN))
!>           The eigenvalues of A, as computed by a partial Schur
!>           decomposition (Z not computed, T only computed as much
!>           as is necessary for determining eigenvalues).  On exit,
!>           W3 contains the eigenvalues of the matrix in A, possibly
!>           perturbed by ZHSEIN.
!>           Modified.
!>
!>  EVECTL - COMPLEX*16 array, dimension (LDU,max(NN))
!>           The conjugate transpose of the (upper triangular) left
!>           eigenvector matrix for the matrix in T1.
!>           Modified.
!>
!>  EVEZTR - COMPLEX*16 array, dimension (LDU,max(NN))
!>           The (upper triangular) right eigenvector matrix for the
!>           matrix in T1.
!>           Modified.
!>
!>  EVECTY - COMPLEX*16 array, dimension (LDU,max(NN))
!>           The conjugate transpose of the left eigenvector matrix
!>           for the matrix in H.
!>           Modified.
!>
!>  EVECTX - COMPLEX*16 array, dimension (LDU,max(NN))
!>           The right eigenvector matrix for the matrix in H.
!>           Modified.
!>
!>  UU     - COMPLEX*16 array, dimension (LDU,max(NN))
!>           Details of the unitary matrix computed by ZGEHRD.
!>           Modified.
!>
!>  TAU    - COMPLEX*16 array, dimension (max(NN))
!>           Further details of the unitary matrix computed by ZGEHRD.
!>           Modified.
!>
!>  WORK   - COMPLEX*16 array, dimension (NWORK)
!>           Workspace.
!>           Modified.
!>
!>  NWORK  - INTEGER
!>           The number of entries in WORK.  NWORK >= 4*NN(j)*NN(j) + 2.
!>
!>  RWORK  - DOUBLE PRECISION array, dimension (max(NN))
!>           Workspace.  Could be equivalenced to IWORK, but not SELECT.
!>           Modified.
!>
!>  IWORK  - INTEGER array, dimension (max(NN))
!>           Workspace.
!>           Modified.
!>
!>  SELECT - LOGICAL array, dimension (max(NN))
!>           Workspace.  Could be equivalenced to IWORK, but not RWORK.
!>           Modified.
!>
!>  RESULT - DOUBLE PRECISION array, dimension (16)
!>           The values computed by the fourteen tests described above.
!>           The values are currently limited to 1/ulp, to avoid
!>           overflow.
!>           Modified.
!>
!>  INFO   - INTEGER
!>           If 0, then everything ran OK.
!>            -1: NSIZES < 0
!>            -2: Some NN(j) < 0
!>            -3: NTYPES < 0
!>            -6: THRESH < 0
!>            -9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ).
!>           -14: LDU < 1 or LDU < NMAX.
!>           -26: NWORK too small.
!>           If  ZLATMR, CLATMS, or CLATME returns an error code, the
!>               absolute value of it is returned.
!>           If 1, then ZHSEQR could not find all the shifts.
!>           If 2, then the EISPACK code (for small blocks) failed.
!>           If >2, then 30*N iterations were not enough to find an
!>               eigenvalue or to decompose the problem.
!>           Modified.
!>
!>-----------------------------------------------------------------------
!>
!>     Some Local Variables and Parameters:
!>     ---- ----- --------- --- ----------
!>
!>     ZERO, ONE       Real 0 and 1.
!>     MAXTYP          The number of types defined.
!>     MTEST           The number of tests defined: care must be taken
!>                     that (1) the size of RESULT, (2) the number of
!>                     tests actually performed, and (3) MTEST agree.
!>     NTEST           The number of tests performed on this matrix
!>                     so far.  This should be less than MTEST, and
!>                     equal to it by the last test.  It will be less
!>                     if any of the routines being tested indicates
!>                     that it could not compute the matrices that
!>                     would be tested.
!>     NMAX            Largest value in NN.
!>     NMATS           The number of matrices generated so far.
!>     NERRS           The number of tests which have exceeded THRESH
!>                     so far (computed by DLAFTS).
!>     COND, CONDS,
!>     IMODE           Values to be passed to the matrix generators.
!>     ANORM           Norm of A; passed to matrix generators.
!>
!>     OVFL, UNFL      Overflow and underflow thresholds.
!>     ULP, ULPINV     Finest relative precision and its inverse.
!>     RTOVFL, RTUNFL,
!>     RTULP, RTULPI   Square roots of the previous 4 values.
!>
!>             The following four arrays decode JTYPE:
!>     KTYPE(j)        The general type (1-10) for type .
!>     KMODE(j)        The MODE value to be passed to the matrix
!>                     generator for type .
!>     KMAGN(j)        The order of magnitude ( O(1),
!>                     O(overflow^(1/2) ), O(underflow^(1/2) )
!>     KCONDS(j)       Selects whether CONDS is to be 1 or
!>                     1/sqrt(ulp).  (0 means irrelevant.)
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **416** of file **zchkhs.f**.

## Author

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