ZCHKST
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
November 2006
Purpose
ZCHKST checks the Hermitian eigenvalue problem routines.
ZHETRD factors A as U S U* , where * means conjugate transpose,
S is real symmetric tridiagonal, and U is unitary.
ZHETRD can use either just the lower or just the upper triangle
of A; ZCHKST 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.
ZHPTRD does the same as ZHETRD, except that A and V are stored
in "packed" format.
ZUNGTR constructs the matrix U from the contents of V and TAU.
ZUPGTR constructs the matrix U from the contents of VP and TAU.
ZSTEQR factors S as Z D1 Z* , where Z is the unitary
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.
DSTERF computes D3, the matrix of eigenvalues, by the
PWK method, which does not yield eigenvectors.
ZPTEQR factors S as Z4 D4 Z4* , for a
Hermitian positive definite tridiagonal matrix.
D5 is the matrix of eigenvalues computed when Z is not
computed.
DSTEBZ 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.
ZSTEIN computes Y, the eigenvectors of S, given the
eigenvalues.
ZSTEDC factors S as Z D1 Z* , where Z is the unitary
matrix of eigenvectors and D1 is a diagonal matrix with
the eigenvalues on the diagonal ('I' option). It may also
update an input unitary matrix, usually the output
from ZHETRD/ZUNGTR or ZHPTRD/ZUPGTR ('V' option). It may
also just compute eigenvalues ('N' option).
ZSTEMR factors S as Z D1 Z* , where Z is the unitary
matrix of eigenvectors and D1 is a diagonal matrix with
the eigenvalues on the diagonal ('I' option). ZSTEMR
uses the Relatively Robust Representation whenever possible.
When ZCHKST 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 Hermitian eigenroutines. For each matrix, a number
of tests will be performed:
(1) | A - V S V* | / ( |A| n ulp ) ZHETRD( UPLO='U', ... )
(2) | I - UV* | / ( n ulp ) ZUNGTR( UPLO='U', ... )
(3) | A - V S V* | / ( |A| n ulp ) ZHETRD( UPLO='L', ... )
(4) | I - UV* | / ( n ulp ) ZUNGTR( UPLO='L', ... )
(5-8) Same as 1-4, but for ZHPTRD and ZUPGTR.
(9) | S - Z D Z* | / ( |S| n ulp ) ZSTEQR('V',...)
(10) | I - ZZ* | / ( n ulp ) ZSTEQR('V',...)
(11) | D1 - D2 | / ( |D1| ulp ) ZSTEQR('N',...)
(12) | D1 - D3 | / ( |D1| ulp ) DSTERF
(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
DSTECH)
For S positive definite,
(14) | S - Z4 D4 Z4* | / ( |S| n ulp ) ZPTEQR('V',...)
(15) | I - Z4 Z4* | / ( n ulp ) ZPTEQR('V',...)
(16) | D4 - D5 | / ( 100 |D4| ulp ) ZPTEQR('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
DSTEBZ( 'A', 'E', ...)
(18) | WA1 - D3 | / ( |D3| ulp ) DSTEBZ( 'A', 'E', ...)
(19) ( max { min | WA2(i)-WA3(j) | } +
i j
max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
i j
DSTEBZ( 'I', 'E', ...)
(20) | S - Y WA1 Y* | / ( |S| n ulp ) DSTEBZ, ZSTEIN
(21) | I - Y Y* | / ( n ulp ) DSTEBZ, ZSTEIN
(22) | S - Z D Z* | / ( |S| n ulp ) ZSTEDC('I')
(23) | I - ZZ* | / ( n ulp ) ZSTEDC('I')
(24) | S - Z D Z* | / ( |S| n ulp ) ZSTEDC('V')
(25) | I - ZZ* | / ( n ulp ) ZSTEDC('V')
(26) | D1 - D2 | / ( |D1| ulp ) ZSTEDC('V') and
ZSTEDC('N')
Test 27 is disabled at the moment because ZSTEMR 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
ZSTEMR('V', 'A')
(28) max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
i
omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
ZSTEMR('V', 'I')
Tests 29 through 34 are disable at present because ZSTEMR
does not handle partial specturm requests.
(29) | S - Z D Z* | / ( |S| n ulp ) ZSTEMR('V', 'I')
(30) | I - ZZ* | / ( n ulp ) ZSTEMR('V', 'I')
(31) ( max { min | WA2(i)-WA3(j) | } +
i j
max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
i j
ZSTEMR('N', 'I') vs. CSTEMR('V', 'I')
(32) | S - Z D Z* | / ( |S| n ulp ) ZSTEMR('V', 'V')
(33) | I - ZZ* | / ( n ulp ) ZSTEMR('V', 'V')
(34) ( max { min | WA2(i)-WA3(j) | } +
i j
max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
i j
ZSTEMR('N', 'V') vs. CSTEMR('V', 'V')
(35) | S - Z D Z* | / ( |S| n ulp ) ZSTEMR('V', 'A')
(36) | I - ZZ* | / ( n ulp ) ZSTEMR('V', 'A')
(37) ( max { min | WA2(i)-WA3(j) | } +
i j
max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
i j
ZSTEMR('N', 'A') vs. CSTEMR('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 unitary 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 unitary 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 unitary 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) Hermitian 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.
ZHETRD factors A as U S U* , where * means conjugate transpose,
S is real symmetric tridiagonal, and U is unitary.
ZHETRD can use either just the lower or just the upper triangle
of A; ZCHKST 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.
ZHPTRD does the same as ZHETRD, except that A and V are stored
in "packed" format.
ZUNGTR constructs the matrix U from the contents of V and TAU.
ZUPGTR constructs the matrix U from the contents of VP and TAU.
ZSTEQR factors S as Z D1 Z* , where Z is the unitary
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.
DSTERF computes D3, the matrix of eigenvalues, by the
PWK method, which does not yield eigenvectors.
ZPTEQR factors S as Z4 D4 Z4* , for a
Hermitian positive definite tridiagonal matrix.
D5 is the matrix of eigenvalues computed when Z is not
computed.
DSTEBZ 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.
ZSTEIN computes Y, the eigenvectors of S, given the
eigenvalues.
ZSTEDC factors S as Z D1 Z* , where Z is the unitary
matrix of eigenvectors and D1 is a diagonal matrix with
the eigenvalues on the diagonal ('I' option). It may also
update an input unitary matrix, usually the output
from ZHETRD/ZUNGTR or ZHPTRD/ZUPGTR ('V' option). It may
also just compute eigenvalues ('N' option).
ZSTEMR factors S as Z D1 Z* , where Z is the unitary
matrix of eigenvectors and D1 is a diagonal matrix with
the eigenvalues on the diagonal ('I' option). ZSTEMR
uses the Relatively Robust Representation whenever possible.
When ZCHKST 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 Hermitian eigenroutines. For each matrix, a number
of tests will be performed:
(1) | A - V S V* | / ( |A| n ulp ) ZHETRD( UPLO='U', ... )
(2) | I - UV* | / ( n ulp ) ZUNGTR( UPLO='U', ... )
(3) | A - V S V* | / ( |A| n ulp ) ZHETRD( UPLO='L', ... )
(4) | I - UV* | / ( n ulp ) ZUNGTR( UPLO='L', ... )
(5-8) Same as 1-4, but for ZHPTRD and ZUPGTR.
(9) | S - Z D Z* | / ( |S| n ulp ) ZSTEQR('V',...)
(10) | I - ZZ* | / ( n ulp ) ZSTEQR('V',...)
(11) | D1 - D2 | / ( |D1| ulp ) ZSTEQR('N',...)
(12) | D1 - D3 | / ( |D1| ulp ) DSTERF
(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
DSTECH)
For S positive definite,
(14) | S - Z4 D4 Z4* | / ( |S| n ulp ) ZPTEQR('V',...)
(15) | I - Z4 Z4* | / ( n ulp ) ZPTEQR('V',...)
(16) | D4 - D5 | / ( 100 |D4| ulp ) ZPTEQR('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
DSTEBZ( 'A', 'E', ...)
(18) | WA1 - D3 | / ( |D3| ulp ) DSTEBZ( 'A', 'E', ...)
(19) ( max { min | WA2(i)-WA3(j) | } +
i j
max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
i j
DSTEBZ( 'I', 'E', ...)
(20) | S - Y WA1 Y* | / ( |S| n ulp ) DSTEBZ, ZSTEIN
(21) | I - Y Y* | / ( n ulp ) DSTEBZ, ZSTEIN
(22) | S - Z D Z* | / ( |S| n ulp ) ZSTEDC('I')
(23) | I - ZZ* | / ( n ulp ) ZSTEDC('I')
(24) | S - Z D Z* | / ( |S| n ulp ) ZSTEDC('V')
(25) | I - ZZ* | / ( n ulp ) ZSTEDC('V')
(26) | D1 - D2 | / ( |D1| ulp ) ZSTEDC('V') and
ZSTEDC('N')
Test 27 is disabled at the moment because ZSTEMR 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
ZSTEMR('V', 'A')
(28) max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
i
omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
ZSTEMR('V', 'I')
Tests 29 through 34 are disable at present because ZSTEMR
does not handle partial specturm requests.
(29) | S - Z D Z* | / ( |S| n ulp ) ZSTEMR('V', 'I')
(30) | I - ZZ* | / ( n ulp ) ZSTEMR('V', 'I')
(31) ( max { min | WA2(i)-WA3(j) | } +
i j
max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
i j
ZSTEMR('N', 'I') vs. CSTEMR('V', 'I')
(32) | S - Z D Z* | / ( |S| n ulp ) ZSTEMR('V', 'V')
(33) | I - ZZ* | / ( n ulp ) ZSTEMR('V', 'V')
(34) ( max { min | WA2(i)-WA3(j) | } +
i j
max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
i j
ZSTEMR('N', 'V') vs. CSTEMR('V', 'V')
(35) | S - Z D Z* | / ( |S| n ulp ) ZSTEMR('V', 'A')
(36) | I - ZZ* | / ( n ulp ) ZSTEMR('V', 'A')
(37) ( max { min | WA2(i)-WA3(j) | } +
i j
max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
i j
ZSTEMR('N', 'A') vs. CSTEMR('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 unitary 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 unitary 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 unitary 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) Hermitian 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.
Arguments
| NSIZES |
(input) INTEGER
The number of sizes of matrices to use. If it is zero,
ZCHKST 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. |
| NTYPES |
(input) INTEGER
The number of elements in DOTYPE. If it is zero, ZCHKST
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 ZCHKST 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/output) COMPLEX*16 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 |
(input) INTEGER
The leading dimension of A. It must be at
least 1 and at least max( NN ). |
| AP |
(workspace) COMPLEX*16 array of
dimension( max(NN)*max(NN+1)/2 )
The matrix A stored in packed format. |
| SD |
(workspace/output) DOUBLE PRECISION array of
dimension( max(NN) )
The diagonal of the tridiagonal matrix computed by ZHETRD. On exit, SD and SE contain the tridiagonal form of the matrix in A. |
| SE |
(workspace/output) DOUBLE PRECISION array of
dimension( max(NN) )
The off-diagonal of the tridiagonal matrix computed by ZHETRD. On exit, SD and SE contain the tridiagonal form of the matrix in A. |
| D1 |
(workspace/output) DOUBLE PRECISION array of
dimension( max(NN) )
The eigenvalues of A, as computed by ZSTEQR simlutaneously with Z. On exit, the eigenvalues in D1 correspond with the matrix in A. |
| D2 |
(workspace/output) DOUBLE PRECISION array of
dimension( max(NN) )
The eigenvalues of A, as computed by ZSTEQR if Z is not computed. On exit, the eigenvalues in D2 correspond with the matrix in A. |
| D3 |
(workspace/output) DOUBLE PRECISION array of
dimension( max(NN) )
The eigenvalues of A, as computed by DSTERF. On exit, the eigenvalues in D3 correspond with the matrix in A. |
| U |
(workspace/output) COMPLEX*16 array of
dimension( LDU, max(NN) ).
The unitary matrix computed by ZHETRD + ZUNGTR. |
| LDU |
(input) INTEGER
The leading dimension of U, Z, and V. It must be at least 1
and at least max( NN ). |
| V |
(workspace/output) COMPLEX*16 array of
dimension( LDU, max(NN) ).
The Housholder vectors computed by ZHETRD 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 ZHETRD, 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 ZUNGTR, set those entries to 1 before using them, and then restore them later.) |
| VP |
(workspace) COMPLEX*16 array of
dimension( max(NN)*max(NN+1)/2 )
The matrix V stored in packed format. |
| TAU |
(workspace/output) COMPLEX*16 array of
dimension( max(NN) )
The Householder factors computed by ZHETRD in reducing A to tridiagonal form. |
| Z |
(workspace/output) COMPLEX*16 array of
dimension( LDU, max(NN) ).
The unitary matrix of eigenvectors computed by ZSTEQR, ZPTEQR, and ZSTEIN. |
| WORK |
(workspace/output) COMPLEX*16 array of
dimension( LWORK )
|
| LWORK |
(input) 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 |
(workspace/output) INTEGER array,
dimension (6 + 6*Nmax + 5 * Nmax * lg Nmax )
where Nmax = max( NN(j), 2 ) and lg = log base 2. Workspace. |
| RWORK |
(workspace/output) DOUBLE PRECISION array of
dimension( ??? )
|
| RESULT |
(output) DOUBLE PRECISION array, dimension (26)
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.
-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 ZLATMR, CLATMS, ZHETRD, ZUNGTR, ZSTEQR, DSTERF, or ZUNMC2 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) ) |