# schkst.f - Man Page

TESTING/EIG/schkst.f

## Synopsis

### Functions/Subroutines

subroutine schkst (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a, lda, ap, sd, se, d1, d2, d3, d4, d5, wa1, wa2, wa3, wr, u, ldu, v, vp, tau, z, work, lwork, iwork, liwork, result, info)
SCHKST

## Function/Subroutine Documentation

### subroutine schkst (integer nsizes, integer, dimension( * ) nn, integer ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed, real thresh, integer nounit, real, dimension( lda, * ) a, integer lda, real, dimension( * ) ap, real, dimension( * ) sd, real, dimension( * ) se, real, dimension( * ) d1, real, dimension( * ) d2, real, dimension( * ) d3, real, dimension( * ) d4, real, dimension( * ) d5, real, dimension( * ) wa1, real, dimension( * ) wa2, real, dimension( * ) wa3, real, dimension( * ) wr, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldu, * ) v, real, dimension( * ) vp, real, dimension( * ) tau, real, dimension( ldu, * ) z, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, real, dimension( * ) result, integer info)

SCHKST

Purpose:

``` SCHKST  checks the symmetric eigenvalue problem routines.

SSYTRD factors A as  U S U' , where ' means transpose,
S is symmetric tridiagonal, and U is orthogonal.
SSYTRD can use either just the lower or just the upper triangle
of A; SCHKST checks both cases.
U is represented as a product of Householder
transformations, whose vectors are stored in the first
n-1 columns of V, and whose scale factors are in TAU.

SSPTRD does the same as SSYTRD, except that A and V are stored
in 'packed' format.

SORGTR constructs the matrix U from the contents of V and TAU.

SOPGTR constructs the matrix U from the contents of VP and TAU.

SSTEQR factors S as  Z D1 Z' , where Z is the orthogonal
matrix of eigenvectors and D1 is a diagonal matrix with
the eigenvalues on the diagonal.  D2 is the matrix of
eigenvalues computed when Z is not computed.

SSTERF computes D3, the matrix of eigenvalues, by the
PWK method, which does not yield eigenvectors.

SPTEQR factors S as  Z4 D4 Z4' , for a
symmetric positive definite tridiagonal matrix.
D5 is the matrix of eigenvalues computed when Z is not
computed.

SSTEBZ computes selected eigenvalues.  WA1, WA2, and
WA3 will denote eigenvalues computed to high
absolute accuracy, with different range options.
WR will denote eigenvalues computed to high relative
accuracy.

SSTEIN computes Y, the eigenvectors of S, given the
eigenvalues.

SSTEDC factors S as Z D1 Z' , where Z is the orthogonal
matrix of eigenvectors and D1 is a diagonal matrix with
the eigenvalues on the diagonal ('I' option). It may also
update an input orthogonal matrix, usually the output
from SSYTRD/SORGTR or SSPTRD/SOPGTR ('V' option). It may
also just compute eigenvalues ('N' option).

SSTEMR factors S as Z D1 Z' , where Z is the orthogonal
matrix of eigenvectors and D1 is a diagonal matrix with
the eigenvalues on the diagonal ('I' option).  SSTEMR
uses the Relatively Robust Representation whenever possible.

When SCHKST is called, a number of matrix 'sizes' ('n's') and a
number of matrix 'types' are specified.  For each size ('n')
and each type of matrix, one matrix will be generated and used
to test the symmetric eigenroutines.  For each matrix, a number
of tests will be performed:

(1)     | A - V S V' | / ( |A| n ulp ) SSYTRD( UPLO='U', ... )

(2)     | I - UV' | / ( n ulp )        SORGTR( UPLO='U', ... )

(3)     | A - V S V' | / ( |A| n ulp ) SSYTRD( UPLO='L', ... )

(4)     | I - UV' | / ( n ulp )        SORGTR( UPLO='L', ... )

(5-8)   Same as 1-4, but for SSPTRD and SOPGTR.

(9)     | S - Z D Z' | / ( |S| n ulp ) SSTEQR('V',...)

(10)    | I - ZZ' | / ( n ulp )        SSTEQR('V',...)

(11)    | D1 - D2 | / ( |D1| ulp )        SSTEQR('N',...)

(12)    | D1 - D3 | / ( |D1| ulp )        SSTERF

(13)    0 if the true eigenvalues (computed by sturm count)
of S are within THRESH of
those in D1.  2*THRESH if they are not.  (Tested using
SSTECH)

For S positive definite,

(14)    | S - Z4 D4 Z4' | / ( |S| n ulp ) SPTEQR('V',...)

(15)    | I - Z4 Z4' | / ( n ulp )        SPTEQR('V',...)

(16)    | D4 - D5 | / ( 100 |D4| ulp )       SPTEQR('N',...)

When S is also diagonally dominant by the factor gamma < 1,

(17)    max | D4(i) - WR(i) | / ( |D4(i)| omega ) ,
i
omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
SSTEBZ( 'A', 'E', ...)

(18)    | WA1 - D3 | / ( |D3| ulp )          SSTEBZ( 'A', 'E', ...)

(19)    ( max { min | WA2(i)-WA3(j) | } +
i     j
max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
i     j
SSTEBZ( 'I', 'E', ...)

(20)    | S - Y WA1 Y' | / ( |S| n ulp )  SSTEBZ, SSTEIN

(21)    | I - Y Y' | / ( n ulp )          SSTEBZ, SSTEIN

(22)    | S - Z D Z' | / ( |S| n ulp )    SSTEDC('I')

(23)    | I - ZZ' | / ( n ulp )           SSTEDC('I')

(24)    | S - Z D Z' | / ( |S| n ulp )    SSTEDC('V')

(25)    | I - ZZ' | / ( n ulp )           SSTEDC('V')

(26)    | D1 - D2 | / ( |D1| ulp )           SSTEDC('V') and
SSTEDC('N')

Test 27 is disabled at the moment because SSTEMR does not
guarantee high relatvie accuracy.

(27)    max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
i
omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
SSTEMR('V', 'A')

(28)    max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
i
omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
SSTEMR('V', 'I')

Tests 29 through 34 are disable at present because SSTEMR
does not handle partial spectrum requests.

(29)    | S - Z D Z' | / ( |S| n ulp )    SSTEMR('V', 'I')

(30)    | I - ZZ' | / ( n ulp )           SSTEMR('V', 'I')

(31)    ( max { min | WA2(i)-WA3(j) | } +
i     j
max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
i     j
SSTEMR('N', 'I') vs. SSTEMR('V', 'I')

(32)    | S - Z D Z' | / ( |S| n ulp )    SSTEMR('V', 'V')

(33)    | I - ZZ' | / ( n ulp )           SSTEMR('V', 'V')

(34)    ( max { min | WA2(i)-WA3(j) | } +
i     j
max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
i     j
SSTEMR('N', 'V') vs. SSTEMR('V', 'V')

(35)    | S - Z D Z' | / ( |S| n ulp )    SSTEMR('V', 'A')

(36)    | I - ZZ' | / ( n ulp )           SSTEMR('V', 'A')

(37)    ( max { min | WA2(i)-WA3(j) | } +
i     j
max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
i     j
SSTEMR('N', 'A') vs. SSTEMR('V', 'A')

The 'sizes' 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)  The zero matrix.
(2)  The identity matrix.

(3)  A diagonal matrix with evenly spaced entries
1, ..., ULP  and random signs.
(ULP = (first number larger than 1) - 1 )
(4)  A diagonal matrix with geometrically spaced entries
1, ..., ULP  and random signs.
(5)  A diagonal matrix with 'clustered' entries 1, ULP, ..., ULP
and random signs.

(6)  Same as (4), but multiplied by SQRT( overflow threshold )
(7)  Same as (4), but multiplied by SQRT( underflow threshold )

(8)  A matrix of the form  U' D U, where U is orthogonal and
D has evenly spaced entries 1, ..., ULP with random signs
on the diagonal.

(9)  A matrix of the form  U' D U, where U is orthogonal and
D has geometrically spaced entries 1, ..., ULP with random
signs on the diagonal.

(10) A matrix of the form  U' D U, where U is orthogonal and
D has 'clustered' entries 1, ULP,..., ULP with random
signs on the diagonal.

(11) Same as (8), but multiplied by SQRT( overflow threshold )
(12) Same as (8), but multiplied by SQRT( underflow threshold )

(13) Symmetric matrix with random entries chosen from (-1,1).
(14) Same as (13), but multiplied by SQRT( overflow threshold )
(15) Same as (13), but multiplied by SQRT( underflow threshold )
(16) Same as (8), but diagonal elements are all positive.
(17) Same as (9), but diagonal elements are all positive.
(18) Same as (10), but diagonal elements are all positive.
(19) Same as (16), but multiplied by SQRT( overflow threshold )
(20) Same as (16), but multiplied by SQRT( underflow threshold )
(21) A diagonally dominant tridiagonal matrix with geometrically
spaced diagonal entries 1, ..., ULP.```
Parameters

NSIZES

```          NSIZES is INTEGER
The number of sizes of matrices to use.  If it is zero,
SCHKST 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, SCHKST
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 is 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

```          ISEED is 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 SCHKST to continue the same random number
sequence.```

THRESH

```          THRESH is REAL
A test will count as 'failed' if the '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

```          NOUNIT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IINFO not equal to 0.)```

A

```          A is REAL array of
dimension ( LDA , max(NN) )
Used to hold the matrix whose eigenvalues are to be
computed.  On exit, A contains the last matrix actually
used.```

LDA

```          LDA is INTEGER
The leading dimension of A.  It must be at
least 1 and at least max( NN ).```

AP

```          AP is REAL array of
dimension( max(NN)*max(NN+1)/2 )
The matrix A stored in packed format.```

SD

```          SD is REAL array of
dimension( max(NN) )
The diagonal of the tridiagonal matrix computed by SSYTRD.
On exit, SD and SE contain the tridiagonal form of the
matrix in A.```

SE

```          SE is REAL array of
dimension( max(NN) )
The off-diagonal of the tridiagonal matrix computed by
SSYTRD.  On exit, SD and SE contain the tridiagonal form of
the matrix in A.```

D1

```          D1 is REAL array of
dimension( max(NN) )
The eigenvalues of A, as computed by SSTEQR simultaneously
with Z.  On exit, the eigenvalues in D1 correspond with the
matrix in A.```

D2

```          D2 is REAL array of
dimension( max(NN) )
The eigenvalues of A, as computed by SSTEQR if Z is not
computed.  On exit, the eigenvalues in D2 correspond with
the matrix in A.```

D3

```          D3 is REAL array of
dimension( max(NN) )
The eigenvalues of A, as computed by SSTERF.  On exit, the
eigenvalues in D3 correspond with the matrix in A.```

D4

```          D4 is REAL array of
dimension( max(NN) )
The eigenvalues of A, as computed by SPTEQR(V).
ZPTEQR factors S as  Z4 D4 Z4*
On exit, the eigenvalues in D4 correspond with the matrix in A.```

D5

```          D5 is REAL array of
dimension( max(NN) )
The eigenvalues of A, as computed by SPTEQR(N)
when Z is not computed. On exit, the
eigenvalues in D4 correspond with the matrix in A.```

WA1

```          WA1 is REAL array of
dimension( max(NN) )
All eigenvalues of A, computed to high
absolute accuracy, with different range options.
as computed by SSTEBZ.```

WA2

```          WA2 is REAL array of
dimension( max(NN) )
Selected eigenvalues of A, computed to high
absolute accuracy, with different range options.
as computed by SSTEBZ.
Choose random values for IL and IU, and ask for the
IL-th through IU-th eigenvalues.```

WA3

```          WA3 is REAL array of
dimension( max(NN) )
Selected eigenvalues of A, computed to high
absolute accuracy, with different range options.
as computed by SSTEBZ.
Determine the values VL and VU of the IL-th and IU-th
eigenvalues and ask for all eigenvalues in this range.```

WR

```          WR is REAL array of
dimension( max(NN) )
All eigenvalues of A, computed to high
absolute accuracy, with different options.
as computed by SSTEBZ.```

U

```          U is REAL array of
dimension( LDU, max(NN) ).
The orthogonal matrix computed by SSYTRD + SORGTR.```

LDU

```          LDU is INTEGER
The leading dimension of U, Z, and V.  It must be at least 1
and at least max( NN ).```

V

```          V is REAL array of
dimension( LDU, max(NN) ).
The Housholder vectors computed by SSYTRD in reducing A to
tridiagonal form.  The vectors computed with UPLO='U' are
in the upper triangle, and the vectors computed with UPLO='L'
are in the lower triangle.  (As described in SSYTRD, the
sub- and superdiagonal are not set to 1, although the
true Householder vector has a 1 in that position.  The
routines that use V, such as SORGTR, set those entries to
1 before using them, and then restore them later.)```

VP

```          VP is REAL array of
dimension( max(NN)*max(NN+1)/2 )
The matrix V stored in packed format.```

TAU

```          TAU is REAL array of
dimension( max(NN) )
The Householder factors computed by SSYTRD in reducing A
to tridiagonal form.```

Z

```          Z is REAL array of
dimension( LDU, max(NN) ).
The orthogonal matrix of eigenvectors computed by SSTEQR,
SPTEQR, and SSTEIN.```

WORK

```          WORK is REAL array of
dimension( LWORK )```

LWORK

```          LWORK is INTEGER
The number of entries in WORK.  This must be at least
1 + 4 * Nmax + 2 * Nmax * lg Nmax + 3 * Nmax**2
where Nmax = max( NN(j), 2 ) and lg = log base 2.```

IWORK

```          IWORK is INTEGER array,
Workspace.```

LIWORK

```          LIWORK is INTEGER
The number of entries in IWORK.  This must be at least
6 + 6*Nmax + 5 * Nmax * lg Nmax
where Nmax = max( NN(j), 2 ) and lg = log base 2.```

RESULT

```          RESULT is REAL array, dimension (26)
The values computed by the tests described above.
The values are currently limited to 1/ulp, to avoid
overflow.```

INFO

```          INFO is INTEGER
If 0, then everything ran OK.
-1: NSIZES < 0
-2: Some NN(j) < 0
-3: NTYPES < 0
-5: THRESH < 0
-9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ).
-23: LDU < 1 or LDU < NMAX.
-29: LWORK too small.
If  SLATMR, SLATMS, SSYTRD, SORGTR, SSTEQR, SSTERF,
or SORMC2 returns an error code, the
absolute value of it is returned.

-----------------------------------------------------------------------

Some Local Variables and Parameters:
---- ----- --------- --- ----------
ZERO, ONE       Real 0 and 1.
MAXTYP          The number of types defined.
NTEST           The number of tests performed, or which can
be performed so far, for the current matrix.
NTESTT          The total number of tests performed so far.
NBLOCK          Blocksize as returned by ENVIR.
NMAX            Largest value in NN.
NMATS           The number of matrices generated so far.
NERRS           The number of tests which have exceeded THRESH
so far.
COND, 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  Square roots of the previous 2 values.
The following four arrays decode JTYPE:
KTYPE(j)        The general type (1-10) for type 'j'.
KMODE(j)        The MODE value to be passed to the matrix
generator for type 'j'.
KMAGN(j)        The order of magnitude ( O(1),
O(overflow^(1/2) ), O(underflow^(1/2) )```
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 587 of file schkst.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page schkst(3) is an alias of schkst.f(3).

Tue Nov 28 2023 12:08:42 Version 3.12.0 LAPACK