DCHKSB
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
November 2006
Purpose
DCHKSB tests the reduction of a symmetric band matrix to tridiagonal
form, used with the symmetric eigenvalue problem.
DSBTRD factors a symmetric band matrix A as U S U' , where ' means
transpose, S is symmetric tridiagonal, and U is orthogonal.
DSBTRD can use either just the lower or just the upper triangle
of A; DCHKSB checks both cases.
When DCHKSB is called, a number of matrix "sizes" ("n's"), a number
of bandwidths ("k's"), and a number of matrix "types" are
specified. For each size ("n"), each bandwidth ("k") less than or
equal to "n", and each type of matrix, one matrix will be generated
and used to test the symmetric banded reduction routine. For each
matrix, a number of tests will be performed:
(1) | A - V S V' | / ( |A| n ulp ) computed by DSBTRD with
UPLO='U'
(2) | I - UU' | / ( n ulp )
(3) | A - V S V' | / ( |A| n ulp ) computed by DSBTRD with
UPLO='L'
(4) | I - UU' | / ( n ulp )
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 )
form, used with the symmetric eigenvalue problem.
DSBTRD factors a symmetric band matrix A as U S U' , where ' means
transpose, S is symmetric tridiagonal, and U is orthogonal.
DSBTRD can use either just the lower or just the upper triangle
of A; DCHKSB checks both cases.
When DCHKSB is called, a number of matrix "sizes" ("n's"), a number
of bandwidths ("k's"), and a number of matrix "types" are
specified. For each size ("n"), each bandwidth ("k") less than or
equal to "n", and each type of matrix, one matrix will be generated
and used to test the symmetric banded reduction routine. For each
matrix, a number of tests will be performed:
(1) | A - V S V' | / ( |A| n ulp ) computed by DSBTRD with
UPLO='U'
(2) | I - UU' | / ( n ulp )
(3) | A - V S V' | / ( |A| n ulp ) computed by DSBTRD with
UPLO='L'
(4) | I - UU' | / ( n ulp )
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 )
Arguments
| NSIZES |
(input) INTEGER
The number of sizes of matrices to use. If it is zero,
DCHKSB does nothing. It must be at least zero. |
| NN |
(input) 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. |
| NWDTHS |
(input) INTEGER
The number of bandwidths to use. If it is zero,
DCHKSB does nothing. It must be at least zero. |
| KK |
(input) INTEGER array, dimension (NWDTHS)
An array containing the bandwidths to be used for the band
matrices. The values must be at least zero. |
| NTYPES |
(input) INTEGER
The number of elements in DOTYPE. If it is zero, DCHKSB
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 |
(input) 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 |
(input/output) 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 DCHKSB to continue the same random number sequence. |
| THRESH |
(input) DOUBLE PRECISION
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 |
(input) INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IINFO not equal to 0.) |
| A |
(input/workspace) DOUBLE PRECISION array, dimension
(LDA, max(NN))
Used to hold the matrix whose eigenvalues are to be computed. |
| LDA |
(input) INTEGER
The leading dimension of A. It must be at least 2 (not 1!)
and at least max( KK )+1. |
| SD |
(workspace) DOUBLE PRECISION array, dimension (max(NN))
Used to hold the diagonal of the tridiagonal matrix computed
by DSBTRD. |
| SE |
(workspace) DOUBLE PRECISION array, dimension (max(NN))
Used to hold the off-diagonal of the tridiagonal matrix
computed by DSBTRD. |
| U |
(workspace) DOUBLE PRECISION array, dimension (LDU, max(NN))
Used to hold the orthogonal matrix computed by DSBTRD.
|
| LDU |
(input) INTEGER
The leading dimension of U. It must be at least 1
and at least max( NN ). |
| WORK |
(workspace) DOUBLE PRECISION array, dimension (LWORK)
|
| LWORK |
(input) INTEGER
The number of entries in WORK. This must be at least
max( LDA+1, max(NN)+1 )*max(NN). |
| RESULT |
(output) DOUBLE PRECISION array, dimension (4)
The values computed by the tests described above.
The values are currently limited to 1/ulp, to avoid overflow. |
| INFO |
(output) INTEGER
If 0, then everything ran OK.
*----------------------------------------------------------------------- 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. 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) ) |