SDRVST
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
November 2006
Purpose
SDRVST checks the symmetric eigenvalue problem drivers.
SSTEV computes all eigenvalues and, optionally,
eigenvectors of a real symmetric tridiagonal matrix.
SSTEVX computes selected eigenvalues and, optionally,
eigenvectors of a real symmetric tridiagonal matrix.
SSTEVR computes selected eigenvalues and, optionally,
eigenvectors of a real symmetric tridiagonal matrix
using the Relatively Robust Representation where it can.
SSYEV computes all eigenvalues and, optionally,
eigenvectors of a real symmetric matrix.
SSYEVX computes selected eigenvalues and, optionally,
eigenvectors of a real symmetric matrix.
SSYEVR computes selected eigenvalues and, optionally,
eigenvectors of a real symmetric matrix
using the Relatively Robust Representation where it can.
SSPEV computes all eigenvalues and, optionally,
eigenvectors of a real symmetric matrix in packed
storage.
SSPEVX computes selected eigenvalues and, optionally,
eigenvectors of a real symmetric matrix in packed
storage.
SSBEV computes all eigenvalues and, optionally,
eigenvectors of a real symmetric band matrix.
SSBEVX computes selected eigenvalues and, optionally,
eigenvectors of a real symmetric band matrix.
SSYEVD computes all eigenvalues and, optionally,
eigenvectors of a real symmetric matrix using
a divide and conquer algorithm.
SSPEVD computes all eigenvalues and, optionally,
eigenvectors of a real symmetric matrix in packed
storage, using a divide and conquer algorithm.
SSBEVD computes all eigenvalues and, optionally,
eigenvectors of a real symmetric band matrix,
using a divide and conquer algorithm.
When SDRVST 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 appropriate drivers. For each matrix and each
driver routine called, the following tests will be performed:
(1) | A - Z D Z' | / ( |A| n ulp )
(2) | I - Z Z' | / ( n ulp )
(3) | D1 - D2 | / ( |D1| ulp )
where Z is the matrix of eigenvectors returned when the
eigenvector option is given and D1 and D2 are the eigenvalues
returned with and without the eigenvector option.
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 eigenvalues
1, ..., ULP and random signs.
(ULP = (first number larger than 1) - 1 )
(4) A diagonal matrix with geometrically spaced eigenvalues
1, ..., ULP and random signs.
(5) A diagonal matrix with "clustered" eigenvalues
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) A band matrix with half bandwidth randomly chosen between
0 and N-1, with evenly spaced eigenvalues 1, ..., ULP
with random signs.
(17) Same as (16), but multiplied by SQRT( overflow threshold )
(18) Same as (16), but multiplied by SQRT( underflow threshold )
SSTEV computes all eigenvalues and, optionally,
eigenvectors of a real symmetric tridiagonal matrix.
SSTEVX computes selected eigenvalues and, optionally,
eigenvectors of a real symmetric tridiagonal matrix.
SSTEVR computes selected eigenvalues and, optionally,
eigenvectors of a real symmetric tridiagonal matrix
using the Relatively Robust Representation where it can.
SSYEV computes all eigenvalues and, optionally,
eigenvectors of a real symmetric matrix.
SSYEVX computes selected eigenvalues and, optionally,
eigenvectors of a real symmetric matrix.
SSYEVR computes selected eigenvalues and, optionally,
eigenvectors of a real symmetric matrix
using the Relatively Robust Representation where it can.
SSPEV computes all eigenvalues and, optionally,
eigenvectors of a real symmetric matrix in packed
storage.
SSPEVX computes selected eigenvalues and, optionally,
eigenvectors of a real symmetric matrix in packed
storage.
SSBEV computes all eigenvalues and, optionally,
eigenvectors of a real symmetric band matrix.
SSBEVX computes selected eigenvalues and, optionally,
eigenvectors of a real symmetric band matrix.
SSYEVD computes all eigenvalues and, optionally,
eigenvectors of a real symmetric matrix using
a divide and conquer algorithm.
SSPEVD computes all eigenvalues and, optionally,
eigenvectors of a real symmetric matrix in packed
storage, using a divide and conquer algorithm.
SSBEVD computes all eigenvalues and, optionally,
eigenvectors of a real symmetric band matrix,
using a divide and conquer algorithm.
When SDRVST 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 appropriate drivers. For each matrix and each
driver routine called, the following tests will be performed:
(1) | A - Z D Z' | / ( |A| n ulp )
(2) | I - Z Z' | / ( n ulp )
(3) | D1 - D2 | / ( |D1| ulp )
where Z is the matrix of eigenvectors returned when the
eigenvector option is given and D1 and D2 are the eigenvalues
returned with and without the eigenvector option.
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 eigenvalues
1, ..., ULP and random signs.
(ULP = (first number larger than 1) - 1 )
(4) A diagonal matrix with geometrically spaced eigenvalues
1, ..., ULP and random signs.
(5) A diagonal matrix with "clustered" eigenvalues
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) A band matrix with half bandwidth randomly chosen between
0 and N-1, with evenly spaced eigenvalues 1, ..., ULP
with random signs.
(17) Same as (16), but multiplied by SQRT( overflow threshold )
(18) Same as (16), but multiplied by SQRT( underflow threshold )
Arguments
| NSIZES |
INTEGER
The number of sizes of matrices to use. If it is zero,
SDRVST 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, SDRVST
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 SDRVST to continue the same random number sequence. Modified. |
| THRESH |
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. 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 |
REAL 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. It must be at
least 1 and at least max( NN ). Not modified. |
| D1 |
REAL array, dimension (max(NN))
The eigenvalues of A, as computed by SSTEQR simlutaneously
with Z. On exit, the eigenvalues in D1 correspond with the matrix in A. Modified. |
| D2 |
REAL array, 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. Modified. |
| D3 |
REAL array, dimension (max(NN))
The eigenvalues of A, as computed by SSTERF. On exit, the
eigenvalues in D3 correspond with the matrix in A. Modified. |
| D4 |
REAL array, dimension
|
| EVEIGS |
REAL array, dimension (max(NN))
The eigenvalues as computed by SSTEV('N', ... )
(I reserve the right to change this to the output of whichever algorithm computes the most accurate eigenvalues). |
| WA1 |
REAL array, dimension
|
| WA2 |
REAL array, dimension
|
| WA3 |
REAL array, dimension
|
| U |
REAL array, dimension (LDU, max(NN))
The orthogonal matrix computed by SSYTRD + SORGTR.
Modified. |
| LDU |
INTEGER
The leading dimension of U, Z, and V. It must be at
least 1 and at least max( NN ). Not modified. |
| V |
REAL array, dimension (LDU, max(NN))
The Housholder vectors computed by SSYTRD in reducing A to
tridiagonal form. Modified. |
| TAU |
REAL array, dimension (max(NN))
The Householder factors computed by SSYTRD in reducing A
to tridiagonal form. Modified. |
| Z |
REAL array, dimension (LDU, max(NN))
The orthogonal matrix of eigenvectors computed by SSTEQR,
SPTEQR, and SSTEIN. Modified. |
| WORK |
REAL array, dimension (LWORK)
Workspace.
Modified. |
| LWORK |
INTEGER
The number of entries in WORK. This must be at least
1 + 4 * Nmax + 2 * Nmax * lg Nmax + 4 * Nmax**2 where Nmax = max( NN(j), 2 ) and lg = log base 2. Not modified. |
| IWORK |
INTEGER array,
dimension (6 + 6*Nmax + 5 * Nmax * lg Nmax )
where Nmax = max( NN(j), 2 ) and lg = log base 2. Workspace. Modified. |
| RESULT |
REAL array, dimension (105)
The values computed by the 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 -5: THRESH < 0 -9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ). -16: LDU < 1 or LDU < NMAX. -21: LWORK too small. If SLATMR, SLATMS, SSYTRD, SORGTR, SSTEQR, SSTERF, or SORMTR returns an error code, the absolute value of it is returned. Modified. *----------------------------------------------------------------------- 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 (computed by SLAFTS). 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) ) The tests performed are: Routine tested 1= | A - U S U' | / ( |A| n ulp ) SSTEV('V', ... ) 2= | I - U U' | / ( n ulp ) SSTEV('V', ... ) 3= |D(with Z) - D(w/o Z)| / (|D| ulp) SSTEV('N', ... ) 4= | A - U S U' | / ( |A| n ulp ) SSTEVX('V','A', ... ) 5= | I - U U' | / ( n ulp ) SSTEVX('V','A', ... ) 6= |D(with Z) - EVEIGS| / (|D| ulp) SSTEVX('N','A', ... ) 7= | A - U S U' | / ( |A| n ulp ) SSTEVR('V','A', ... ) 8= | I - U U' | / ( n ulp ) SSTEVR('V','A', ... ) 9= |D(with Z) - EVEIGS| / (|D| ulp) SSTEVR('N','A', ... ) 10= | A - U S U' | / ( |A| n ulp ) SSTEVX('V','I', ... ) 11= | I - U U' | / ( n ulp ) SSTEVX('V','I', ... ) 12= |D(with Z) - D(w/o Z)| / (|D| ulp) SSTEVX('N','I', ... ) 13= | A - U S U' | / ( |A| n ulp ) SSTEVX('V','V', ... ) 14= | I - U U' | / ( n ulp ) SSTEVX('V','V', ... ) 15= |D(with Z) - D(w/o Z)| / (|D| ulp) SSTEVX('N','V', ... ) 16= | A - U S U' | / ( |A| n ulp ) SSTEVD('V', ... ) 17= | I - U U' | / ( n ulp ) SSTEVD('V', ... ) 18= |D(with Z) - EVEIGS| / (|D| ulp) SSTEVD('N', ... ) 19= | A - U S U' | / ( |A| n ulp ) SSTEVR('V','I', ... ) 20= | I - U U' | / ( n ulp ) SSTEVR('V','I', ... ) 21= |D(with Z) - D(w/o Z)| / (|D| ulp) SSTEVR('N','I', ... ) 22= | A - U S U' | / ( |A| n ulp ) SSTEVR('V','V', ... ) 23= | I - U U' | / ( n ulp ) SSTEVR('V','V', ... ) 24= |D(with Z) - D(w/o Z)| / (|D| ulp) SSTEVR('N','V', ... ) 25= | A - U S U' | / ( |A| n ulp ) SSYEV('L','V', ... ) 26= | I - U U' | / ( n ulp ) SSYEV('L','V', ... ) 27= |D(with Z) - D(w/o Z)| / (|D| ulp) SSYEV('L','N', ... ) 28= | A - U S U' | / ( |A| n ulp ) SSYEVX('L','V','A', ... ) 29= | I - U U' | / ( n ulp ) SSYEVX('L','V','A', ... ) 30= |D(with Z) - D(w/o Z)| / (|D| ulp) SSYEVX('L','N','A', ... ) 31= | A - U S U' | / ( |A| n ulp ) SSYEVX('L','V','I', ... ) 32= | I - U U' | / ( n ulp ) SSYEVX('L','V','I', ... ) 33= |D(with Z) - D(w/o Z)| / (|D| ulp) SSYEVX('L','N','I', ... ) 34= | A - U S U' | / ( |A| n ulp ) SSYEVX('L','V','V', ... ) 35= | I - U U' | / ( n ulp ) SSYEVX('L','V','V', ... ) 36= |D(with Z) - D(w/o Z)| / (|D| ulp) SSYEVX('L','N','V', ... ) 37= | A - U S U' | / ( |A| n ulp ) SSPEV('L','V', ... ) 38= | I - U U' | / ( n ulp ) SSPEV('L','V', ... ) 39= |D(with Z) - D(w/o Z)| / (|D| ulp) SSPEV('L','N', ... ) 40= | A - U S U' | / ( |A| n ulp ) SSPEVX('L','V','A', ... ) 41= | I - U U' | / ( n ulp ) SSPEVX('L','V','A', ... ) 42= |D(with Z) - D(w/o Z)| / (|D| ulp) SSPEVX('L','N','A', ... ) 43= | A - U S U' | / ( |A| n ulp ) SSPEVX('L','V','I', ... ) 44= | I - U U' | / ( n ulp ) SSPEVX('L','V','I', ... ) 45= |D(with Z) - D(w/o Z)| / (|D| ulp) SSPEVX('L','N','I', ... ) 46= | A - U S U' | / ( |A| n ulp ) SSPEVX('L','V','V', ... ) 47= | I - U U' | / ( n ulp ) SSPEVX('L','V','V', ... ) 48= |D(with Z) - D(w/o Z)| / (|D| ulp) SSPEVX('L','N','V', ... ) 49= | A - U S U' | / ( |A| n ulp ) SSBEV('L','V', ... ) 50= | I - U U' | / ( n ulp ) SSBEV('L','V', ... ) 51= |D(with Z) - D(w/o Z)| / (|D| ulp) SSBEV('L','N', ... ) 52= | A - U S U' | / ( |A| n ulp ) SSBEVX('L','V','A', ... ) 53= | I - U U' | / ( n ulp ) SSBEVX('L','V','A', ... ) 54= |D(with Z) - D(w/o Z)| / (|D| ulp) SSBEVX('L','N','A', ... ) 55= | A - U S U' | / ( |A| n ulp ) SSBEVX('L','V','I', ... ) 56= | I - U U' | / ( n ulp ) SSBEVX('L','V','I', ... ) 57= |D(with Z) - D(w/o Z)| / (|D| ulp) SSBEVX('L','N','I', ... ) 58= | A - U S U' | / ( |A| n ulp ) SSBEVX('L','V','V', ... ) 59= | I - U U' | / ( n ulp ) SSBEVX('L','V','V', ... ) 60= |D(with Z) - D(w/o Z)| / (|D| ulp) SSBEVX('L','N','V', ... ) 61= | A - U S U' | / ( |A| n ulp ) SSYEVD('L','V', ... ) 62= | I - U U' | / ( n ulp ) SSYEVD('L','V', ... ) 63= |D(with Z) - D(w/o Z)| / (|D| ulp) SSYEVD('L','N', ... ) 64= | A - U S U' | / ( |A| n ulp ) SSPEVD('L','V', ... ) 65= | I - U U' | / ( n ulp ) SSPEVD('L','V', ... ) 66= |D(with Z) - D(w/o Z)| / (|D| ulp) SSPEVD('L','N', ... ) 67= | A - U S U' | / ( |A| n ulp ) SSBEVD('L','V', ... ) 68= | I - U U' | / ( n ulp ) SSBEVD('L','V', ... ) 69= |D(with Z) - D(w/o Z)| / (|D| ulp) SSBEVD('L','N', ... ) 70= | A - U S U' | / ( |A| n ulp ) SSYEVR('L','V','A', ... ) 71= | I - U U' | / ( n ulp ) SSYEVR('L','V','A', ... ) 72= |D(with Z) - D(w/o Z)| / (|D| ulp) SSYEVR('L','N','A', ... ) 73= | A - U S U' | / ( |A| n ulp ) SSYEVR('L','V','I', ... ) 74= | I - U U' | / ( n ulp ) SSYEVR('L','V','I', ... ) 75= |D(with Z) - D(w/o Z)| / (|D| ulp) SSYEVR('L','N','I', ... ) 76= | A - U S U' | / ( |A| n ulp ) SSYEVR('L','V','V', ... ) 77= | I - U U' | / ( n ulp ) SSYEVR('L','V','V', ... ) 78= |D(with Z) - D(w/o Z)| / (|D| ulp) SSYEVR('L','N','V', ... ) Tests 25 through 78 are repeated (as tests 79 through 132) with UPLO='U' To be added in 1999 79= | A - U S U' | / ( |A| n ulp ) SSPEVR('L','V','A', ... ) 80= | I - U U' | / ( n ulp ) SSPEVR('L','V','A', ... ) 81= |D(with Z) - D(w/o Z)| / (|D| ulp) SSPEVR('L','N','A', ... ) 82= | A - U S U' | / ( |A| n ulp ) SSPEVR('L','V','I', ... ) 83= | I - U U' | / ( n ulp ) SSPEVR('L','V','I', ... ) 84= |D(with Z) - D(w/o Z)| / (|D| ulp) SSPEVR('L','N','I', ... ) 85= | A - U S U' | / ( |A| n ulp ) SSPEVR('L','V','V', ... ) 86= | I - U U' | / ( n ulp ) SSPEVR('L','V','V', ... ) 87= |D(with Z) - D(w/o Z)| / (|D| ulp) SSPEVR('L','N','V', ... ) 88= | A - U S U' | / ( |A| n ulp ) SSBEVR('L','V','A', ... ) 89= | I - U U' | / ( n ulp ) SSBEVR('L','V','A', ... ) 90= |D(with Z) - D(w/o Z)| / (|D| ulp) SSBEVR('L','N','A', ... ) 91= | A - U S U' | / ( |A| n ulp ) SSBEVR('L','V','I', ... ) 92= | I - U U' | / ( n ulp ) SSBEVR('L','V','I', ... ) 93= |D(with Z) - D(w/o Z)| / (|D| ulp) SSBEVR('L','N','I', ... ) 94= | A - U S U' | / ( |A| n ulp ) SSBEVR('L','V','V', ... ) 95= | I - U U' | / ( n ulp ) SSBEVR('L','V','V', ... ) 96= |D(with Z) - D(w/o Z)| / (|D| ulp) SSBEVR('L','N','V', ... ) |