.TH "TESTING/EIG/zdrges3.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME TESTING/EIG/zdrges3.f .SH SYNOPSIS .br .PP .SS "Functions/Subroutines" .in +1c .ti -1c .RI "subroutine \fBzdrges3\fP (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a, lda, b, s, t, q, ldq, z, alpha, beta, work, lwork, rwork, result, bwork, info)" .br .RI "\fBZDRGES3\fP " .in -1c .SH "Function/Subroutine Documentation" .PP .SS "subroutine zdrges3 (integer nsizes, integer, dimension( * ) nn, integer ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed, double precision thresh, integer nounit, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( lda, * ) b, complex*16, dimension( lda, * ) s, complex*16, dimension( lda, * ) t, complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( ldq, * ) z, complex*16, dimension( * ) alpha, complex*16, dimension( * ) beta, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, double precision, dimension( 13 ) result, logical, dimension( * ) bwork, integer info)" .PP \fBZDRGES3\fP .PP \fBPurpose:\fP .RS 4 .PP .nf !> !> ZDRGES3 checks the nonsymmetric generalized eigenvalue (Schur form) !> problem driver ZGGES3\&. !> !> ZGGES3 factors A and B as Q*S*Z' and Q*T*Z' , where ' means conjugate !> transpose, S and T are upper triangular (i\&.e\&., in generalized Schur !> form), and Q and Z are unitary\&. It also computes the generalized !> eigenvalues (alpha(j),beta(j)), j=1,\&.\&.\&.,n\&. Thus, !> w(j) = alpha(j)/beta(j) is a root of the characteristic equation !> !> det( A - w(j) B ) = 0 !> !> Optionally it also reorder the eigenvalues so that a selected !> cluster of eigenvalues appears in the leading diagonal block of the !> Schur forms\&. !> !> When ZDRGES3 is called, a number of matrix () and a !> number of matrix are specified\&. For each size () !> and each TYPE of matrix, a pair of matrices (A, B) will be generated !> and used for testing\&. For each matrix pair, the following 13 tests !> will be performed and compared with the threshold THRESH except !> the tests (5), (11) and (13)\&. !> !> !> (1) | A - Q S Z' | / ( |A| n ulp ) (no sorting of eigenvalues) !> !> !> (2) | B - Q T Z' | / ( |B| n ulp ) (no sorting of eigenvalues) !> !> !> (3) | I - QQ' | / ( n ulp ) (no sorting of eigenvalues) !> !> !> (4) | I - ZZ' | / ( n ulp ) (no sorting of eigenvalues) !> !> (5) if A is in Schur form (i\&.e\&. triangular form) (no sorting of !> eigenvalues) !> !> (6) if eigenvalues = diagonal elements of the Schur form (S, T), !> i\&.e\&., test the maximum over j of D(j) where: !> !> |alpha(j) - S(j,j)| |beta(j) - T(j,j)| !> D(j) = ------------------------ + ----------------------- !> max(|alpha(j)|,|S(j,j)|) max(|beta(j)|,|T(j,j)|) !> !> (no sorting of eigenvalues) !> !> (7) | (A,B) - Q (S,T) Z' | / ( |(A,B)| n ulp ) !> (with sorting of eigenvalues)\&. !> !> (8) | I - QQ' | / ( n ulp ) (with sorting of eigenvalues)\&. !> !> (9) | I - ZZ' | / ( n ulp ) (with sorting of eigenvalues)\&. !> !> (10) if A is in Schur form (i\&.e\&. quasi-triangular form) !> (with sorting of eigenvalues)\&. !> !> (11) if eigenvalues = diagonal elements of the Schur form (S, T), !> i\&.e\&. test the maximum over j of D(j) where: !> !> |alpha(j) - S(j,j)| |beta(j) - T(j,j)| !> D(j) = ------------------------ + ----------------------- !> max(|alpha(j)|,|S(j,j)|) max(|beta(j)|,|T(j,j)|) !> !> (with sorting of eigenvalues)\&. !> !> (12) if sorting worked and SDIM is the number of eigenvalues !> which were CELECTed\&. !> !> Test Matrices !> ============= !> !> The sizes of the test matrices are specified by an array !> NN(1:NSIZES); the value of each element NN(j) specifies one size\&. !> The are specified by a logical array DOTYPE( 1:NTYPES ); if !> DOTYPE(j) is \&.TRUE\&., then matrix type will be generated\&. !> Currently, the list of possible types is: !> !> (1) ( 0, 0 ) (a pair of zero matrices) !> !> (2) ( I, 0 ) (an identity and a zero matrix) !> !> (3) ( 0, I ) (an identity and a zero matrix) !> !> (4) ( I, I ) (a pair of identity matrices) !> !> t t !> (5) ( J , J ) (a pair of transposed Jordan blocks) !> !> t ( I 0 ) !> (6) ( X, Y ) where X = ( J 0 ) and Y = ( t ) !> ( 0 I ) ( 0 J ) !> and I is a k x k identity and J a (k+1)x(k+1) !> Jordan block; k=(N-1)/2 !> !> (7) ( D, I ) where D is diag( 0, 1,\&.\&.\&., N-1 ) (a diagonal !> matrix with those diagonal entries\&.) !> (8) ( I, D ) !> !> (9) ( big*D, small*I ) where is near overflow and small=1/big !> !> (10) ( small*D, big*I ) !> !> (11) ( big*I, small*D ) !> !> (12) ( small*I, big*D ) !> !> (13) ( big*D, big*I ) !> !> (14) ( small*D, small*I ) !> !> (15) ( D1, D2 ) where D1 is diag( 0, 0, 1, \&.\&.\&., N-3, 0 ) and !> D2 is diag( 0, N-3, N-4,\&.\&.\&., 1, 0, 0 ) !> t t !> (16) Q ( J , J ) Z where Q and Z are random orthogonal matrices\&. !> !> (17) Q ( T1, T2 ) Z where T1 and T2 are upper triangular matrices !> with random O(1) entries above the diagonal !> and diagonal entries diag(T1) = !> ( 0, 0, 1, \&.\&.\&., N-3, 0 ) and diag(T2) = !> ( 0, N-3, N-4,\&.\&.\&., 1, 0, 0 ) !> !> (18) Q ( T1, T2 ) Z diag(T1) = ( 0, 0, 1, 1, s, \&.\&.\&., s, 0 ) !> diag(T2) = ( 0, 1, 0, 1,\&.\&.\&., 1, 0 ) !> s = machine precision\&. !> !> (19) Q ( T1, T2 ) Z diag(T1)=( 0,0,1,1, 1-d, \&.\&.\&., 1-(N-5)*d=s, 0 ) !> diag(T2) = ( 0, 1, 0, 1, \&.\&.\&., 1, 0 ) !> !> N-5 !> (20) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, 1, a, \&.\&.\&., a =s, 0 ) !> diag(T2) = ( 0, 1, 0, 1, \&.\&.\&., 1, 0, 0 ) !> !> (21) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, r1, r2, \&.\&.\&., r(N-4), 0 ) !> diag(T2) = ( 0, 1, 0, 1, \&.\&.\&., 1, 0, 0 ) !> where r1,\&.\&.\&., r(N-4) are random\&. !> !> (22) Q ( big*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, \&.\&.\&., N-3, 0 ) !> diag(T2) = ( 0, 1, \&.\&.\&., 1, 0, 0 ) !> !> (23) Q ( small*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, \&.\&.\&., N-3, 0 ) !> diag(T2) = ( 0, 1, \&.\&.\&., 1, 0, 0 ) !> !> (24) Q ( small*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, \&.\&.\&., N-3, 0 ) !> diag(T2) = ( 0, 1, \&.\&.\&., 1, 0, 0 ) !> !> (25) Q ( big*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, \&.\&.\&., N-3, 0 ) !> diag(T2) = ( 0, 1, \&.\&.\&., 1, 0, 0 ) !> !> (26) Q ( T1, T2 ) Z where T1 and T2 are random upper-triangular !> matrices\&. !> !> .fi .PP .RE .PP \fBParameters\fP .RS 4 \fINSIZES\fP .PP .nf !> NSIZES is INTEGER !> The number of sizes of matrices to use\&. If it is zero, !> DDRGES3 does nothing\&. NSIZES >= 0\&. !> .fi .PP .br \fINN\fP .PP .nf !> NN is INTEGER array, dimension (NSIZES) !> An array containing the sizes to be used for the matrices\&. !> Zero values will be skipped\&. NN >= 0\&. !> .fi .PP .br \fINTYPES\fP .PP .nf !> NTYPES is INTEGER !> The number of elements in DOTYPE\&. If it is zero, DDRGES3 !> 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 on input\&. !> This is only useful if DOTYPE(1:MAXTYP) is \&.FALSE\&. and !> DOTYPE(MAXTYP+1) is \&.TRUE\&. \&. !> .fi .PP .br \fIDOTYPE\fP .PP .nf !> 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\&. !> .fi .PP .br \fIISEED\fP .PP .nf !> 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 DDRGES3 to continue the same random number !> sequence\&. !> .fi .PP .br \fITHRESH\fP .PP .nf !> THRESH is DOUBLE PRECISION !> A test will count as if the , 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\&. THRESH >= 0\&. !> .fi .PP .br \fINOUNIT\fP .PP .nf !> NOUNIT is INTEGER !> The FORTRAN unit number for printing out error messages !> (e\&.g\&., if a routine returns IINFO not equal to 0\&.) !> .fi .PP .br \fIA\fP .PP .nf !> A is COMPLEX*16 array, dimension(LDA, max(NN)) !> Used to hold the original A matrix\&. Used as input only !> if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=\&.FALSE\&., and !> DOTYPE(MAXTYP+1)=\&.TRUE\&. !> .fi .PP .br \fILDA\fP .PP .nf !> LDA is INTEGER !> The leading dimension of A, B, S, and T\&. !> It must be at least 1 and at least max( NN )\&. !> .fi .PP .br \fIB\fP .PP .nf !> B is COMPLEX*16 array, dimension(LDA, max(NN)) !> Used to hold the original B matrix\&. Used as input only !> if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=\&.FALSE\&., and !> DOTYPE(MAXTYP+1)=\&.TRUE\&. !> .fi .PP .br \fIS\fP .PP .nf !> S is COMPLEX*16 array, dimension (LDA, max(NN)) !> The Schur form matrix computed from A by ZGGES3\&. On exit, S !> contains the Schur form matrix corresponding to the matrix !> in A\&. !> .fi .PP .br \fIT\fP .PP .nf !> T is COMPLEX*16 array, dimension (LDA, max(NN)) !> The upper triangular matrix computed from B by ZGGES3\&. !> .fi .PP .br \fIQ\fP .PP .nf !> Q is COMPLEX*16 array, dimension (LDQ, max(NN)) !> The (left) orthogonal matrix computed by ZGGES3\&. !> .fi .PP .br \fILDQ\fP .PP .nf !> LDQ is INTEGER !> The leading dimension of Q and Z\&. It must !> be at least 1 and at least max( NN )\&. !> .fi .PP .br \fIZ\fP .PP .nf !> Z is COMPLEX*16 array, dimension( LDQ, max(NN) ) !> The (right) orthogonal matrix computed by ZGGES3\&. !> .fi .PP .br \fIALPHA\fP .PP .nf !> ALPHA is COMPLEX*16 array, dimension (max(NN)) !> .fi .PP .br \fIBETA\fP .PP .nf !> BETA is COMPLEX*16 array, dimension (max(NN)) !> !> The generalized eigenvalues of (A,B) computed by ZGGES3\&. !> ALPHA(k) / BETA(k) is the k-th generalized eigenvalue of A !> and B\&. !> .fi .PP .br \fIWORK\fP .PP .nf !> WORK is COMPLEX*16 array, dimension (LWORK) !> .fi .PP .br \fILWORK\fP .PP .nf !> LWORK is INTEGER !> The dimension of the array WORK\&. LWORK >= 3*N*N\&. !> .fi .PP .br \fIRWORK\fP .PP .nf !> RWORK is DOUBLE PRECISION array, dimension ( 8*N ) !> Real workspace\&. !> .fi .PP .br \fIRESULT\fP .PP .nf !> RESULT is DOUBLE PRECISION array, dimension (15) !> The values computed by the tests described above\&. !> The values are currently limited to 1/ulp, to avoid overflow\&. !> .fi .PP .br \fIBWORK\fP .PP .nf !> BWORK is LOGICAL array, dimension (N) !> .fi .PP .br \fIINFO\fP .PP .nf !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value\&. !> > 0: A routine returned an error code\&. INFO is the !> absolute value of the INFO value returned\&. !> .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .PP Definition at line \fB378\fP of file \fBzdrges3\&.f\fP\&. .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.