# schksb2stg.f - Man Page

TESTING/EIG/schksb2stg.f

## Synopsis

### Functions/Subroutines

subroutine **schksb2stg** (nsizes, nn, nwdths, kk, ntypes, dotype, iseed, thresh, nounit, a, lda, sd, se, d1, d2, d3, u, ldu, work, lwork, result, info)**SCHKSB2STG**

## Function/Subroutine Documentation

### subroutine schksb2stg (integer nsizes, integer, dimension( * ) nn, integer nwdths, integer, dimension( * ) kk, integer ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed, real thresh, integer nounit, real, dimension( lda, * ) a, integer lda, real, dimension( * ) sd, real, dimension( * ) se, real, dimension( * ) d1, real, dimension( * ) d2, real, dimension( * ) d3, real, dimension( ldu, * ) u, integer ldu, real, dimension( * ) work, integer lwork, real, dimension( * ) result, integer info)

**SCHKSB2STG**

**Purpose:**

SCHKSB2STG tests the reduction of a symmetric band matrix to tridiagonal form, used with the symmetric eigenvalue problem. SSBTRD factors a symmetric band matrix A as U S U' , where ' means transpose, S is symmetric tridiagonal, and U is orthogonal. SSBTRD can use either just the lower or just the upper triangle of A; SCHKSB2STG checks both cases. SSYTRD_SB2ST factors a symmetric band matrix A as U S U' , where ' means transpose, S is symmetric tridiagonal, and U is orthogonal. SSYTRD_SB2ST can use either just the lower or just the upper triangle of A; SCHKSB2STG checks both cases. SSTEQR factors S as Z D1 Z'. D1 is the matrix of eigenvalues computed when Z is not computed and from the S resulting of SSBTRD 'U' (used as reference for SSYTRD_SB2ST) D2 is the matrix of eigenvalues computed when Z is not computed and from the S resulting of SSYTRD_SB2ST 'U'. D3 is the matrix of eigenvalues computed when Z is not computed and from the S resulting of SSYTRD_SB2ST 'L'. When SCHKSB2STG 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 SSBTRD with UPLO='U' (2) | I - UU' | / ( n ulp ) (3) | A - V S V' | / ( |A| n ulp ) computed by SSBTRD with UPLO='L' (4) | I - UU' | / ( n ulp ) (5) | D1 - D2 | / ( |D1| ulp ) where D1 is computed by SSBTRD with UPLO='U' and D2 is computed by SSYTRD_SB2ST with UPLO='U' (6) | D1 - D3 | / ( |D1| ulp ) where D1 is computed by SSBTRD with UPLO='U' and D3 is computed by SSYTRD_SB2ST with UPLO='L' 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 )

**Parameters***NSIZES*NSIZES is INTEGER The number of sizes of matrices to use. If it is zero, SCHKSB2STG 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.

*NWDTHS*NWDTHS is INTEGER The number of bandwidths to use. If it is zero, SCHKSB2STG does nothing. It must be at least zero.

*KK*KK is INTEGER array, dimension (NWDTHS) An array containing the bandwidths to be used for the band matrices. The values must be at least zero.

*NTYPES*NTYPES is INTEGER The number of elements in DOTYPE. If it is zero, SCHKSB2STG 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 SCHKSB2STG 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, dimension (LDA, max(NN)) Used to hold the matrix whose eigenvalues are to be computed.

*LDA*LDA is INTEGER The leading dimension of A. It must be at least 2 (not 1!) and at least max( KK )+1.

*SD*SD is REAL array, dimension (max(NN)) Used to hold the diagonal of the tridiagonal matrix computed by SSBTRD.

*SE*SE is REAL array, dimension (max(NN)) Used to hold the off-diagonal of the tridiagonal matrix computed by SSBTRD.

*D1*D1 is REAL array, dimension (max(NN))

*D2*D2 is REAL array, dimension (max(NN))

*D3*D3 is REAL array, dimension (max(NN))

*U*U is REAL array, dimension (LDU, max(NN)) Used to hold the orthogonal matrix computed by SSBTRD.

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

*WORK*WORK is REAL array, dimension (LWORK)

*LWORK*LWORK is INTEGER The number of entries in WORK. This must be at least max( LDA+1, max(NN)+1 )*max(NN).

*RESULT*RESULT is REAL array, dimension (4) 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. ----------------------------------------------------------------------- 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) )

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **329** of file **schksb2stg.f**.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page schksb2stg(3) is an alias of schksb2stg.f(3).

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