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java.lang.Object | +----org.netlib.lapack.DHGEQZ
DHGEQZ is a simplified interface to the JLAPACK routine dhgeqz. This interface converts Java-style 2D row-major arrays into the 1D column-major linearized arrays expected by the lower level JLAPACK routines. Using this interface also allows you to omit offset and leading dimension arguments. However, because of these conversions, these routines will be slower than the low level ones. Following is the description from the original Fortran source. Contact seymour@cs.utk.edu with any questions.* .. * * Purpose * ======= * * DHGEQZ implements a single-/double-shift version of the QZ method for * finding the generalized eigenvalues * * w(j)=(ALPHAR(j) + i*ALPHAI(j))/BETAR(j) of the equation * * det( A - w(i) B ) = 0 * * In addition, the pair A,B may be reduced to generalized Schur form: * B is upper triangular, and A is block upper triangular, where the * diagonal blocks are either 1-by-1 or 2-by-2, the 2-by-2 blocks having * complex generalized eigenvalues (see the description of the argument * JOB.) * * If JOB='S', then the pair (A,B) is simultaneously reduced to Schur * form by applying one orthogonal tranformation (usually called Q) on * the left and another (usually called Z) on the right. The 2-by-2 * upper-triangular diagonal blocks of B corresponding to 2-by-2 blocks * of A will be reduced to positive diagonal matrices. (I.e., * if A(j+1,j) is non-zero, then B(j+1,j)=B(j,j+1)=0 and B(j,j) and * B(j+1,j+1) will be positive.) * * If JOB='E', then at each iteration, the same transformations * are computed, but they are only applied to those parts of A and B * which are needed to compute ALPHAR, ALPHAI, and BETAR. * * If JOB='S' and COMPQ and COMPZ are 'V' or 'I', then the orthogonal * transformations used to reduce (A,B) are accumulated into the arrays * Q and Z s.t.: * * Q(in) A(in) Z(in)* = Q(out) A(out) Z(out)* * Q(in) B(in) Z(in)* = Q(out) B(out) Z(out)* * * Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix * Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973), * pp. 241--256. * * Arguments * ========= * * JOB (input) CHARACTER*1 * = 'E': compute only ALPHAR, ALPHAI, and BETA. A and B will * not necessarily be put into generalized Schur form. * = 'S': put A and B into generalized Schur form, as well * as computing ALPHAR, ALPHAI, and BETA. * * COMPQ (input) CHARACTER*1 * = 'N': do not modify Q. * = 'V': multiply the array Q on the right by the transpose of * the orthogonal tranformation that is applied to the * left side of A and B to reduce them to Schur form. * = 'I': like COMPQ='V', except that Q will be initialized to * the identity first. * * COMPZ (input) CHARACTER*1 * = 'N': do not modify Z. * = 'V': multiply the array Z on the right by the orthogonal * tranformation that is applied to the right side of * A and B to reduce them to Schur form. * = 'I': like COMPZ='V', except that Z will be initialized to * the identity first. * * N (input) INTEGER * The order of the matrices A, B, Q, and Z. N >= 0. * * ILO (input) INTEGER * IHI (input) INTEGER * It is assumed that A is already upper triangular in rows and * columns 1:ILO-1 and IHI+1:N. * 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. * * A (input/output) DOUBLE PRECISION array, dimension (LDA, N) * On entry, the N-by-N upper Hessenberg matrix A. Elements * below the subdiagonal must be zero. * If JOB='S', then on exit A and B will have been * simultaneously reduced to generalized Schur form. * If JOB='E', then on exit A will have been destroyed. * The diagonal blocks will be correct, but the off-diagonal * portion will be meaningless. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max( 1, N ). * * B (input/output) DOUBLE PRECISION array, dimension (LDB, N) * On entry, the N-by-N upper triangular matrix B. Elements * below the diagonal must be zero. 2-by-2 blocks in B * corresponding to 2-by-2 blocks in A will be reduced to * positive diagonal form. (I.e., if A(j+1,j) is non-zero, * then B(j+1,j)=B(j,j+1)=0 and B(j,j) and B(j+1,j+1) will be * positive.) * If JOB='S', then on exit A and B will have been * simultaneously reduced to Schur form. * If JOB='E', then on exit B will have been destroyed. * Elements corresponding to diagonal blocks of A will be * correct, but the off-diagonal portion will be meaningless. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max( 1, N ). * * ALPHAR (output) DOUBLE PRECISION array, dimension (N) * ALPHAR(1:N) will be set to real parts of the diagonal * elements of A that would result from reducing A and B to * Schur form and then further reducing them both to triangular * form using unitary transformations s.t. the diagonal of B * was non-negative real. Thus, if A(j,j) is in a 1-by-1 block * (i.e., A(j+1,j)=A(j,j+1)=0), then ALPHAR(j)=A(j,j). * Note that the (real or complex) values * (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the * generalized eigenvalues of the matrix pencil A - wB. * * ALPHAI (output) DOUBLE PRECISION array, dimension (N) * ALPHAI(1:N) will be set to imaginary parts of the diagonal * elements of A that would result from reducing A and B to * Schur form and then further reducing them both to triangular * form using unitary transformations s.t. the diagonal of B * was non-negative real. Thus, if A(j,j) is in a 1-by-1 block * (i.e., A(j+1,j)=A(j,j+1)=0), then ALPHAR(j)=0. * Note that the (real or complex) values * (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the * generalized eigenvalues of the matrix pencil A - wB. * * BETA (output) DOUBLE PRECISION array, dimension (N) * BETA(1:N) will be set to the (real) diagonal elements of B * that would result from reducing A and B to Schur form and * then further reducing them both to triangular form using * unitary transformations s.t. the diagonal of B was * non-negative real. Thus, if A(j,j) is in a 1-by-1 block * (i.e., A(j+1,j)=A(j,j+1)=0), then BETA(j)=B(j,j). * Note that the (real or complex) values * (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the * generalized eigenvalues of the matrix pencil A - wB. * (Note that BETA(1:N) will always be non-negative, and no * BETAI is necessary.) * * Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N) * If COMPQ='N', then Q will not be referenced. * If COMPQ='V' or 'I', then the transpose of the orthogonal * transformations which are applied to A and B on the left * will be applied to the array Q on the right. * * LDQ (input) INTEGER * The leading dimension of the array Q. LDQ >= 1. * If COMPQ='V' or 'I', then LDQ >= N. * * Z (input/output) DOUBLE PRECISION array, dimension (LDZ, N) * If COMPZ='N', then Z will not be referenced. * If COMPZ='V' or 'I', then the orthogonal transformations * which are applied to A and B on the right will be applied * to the array Z on the right. * * LDZ (input) INTEGER * The leading dimension of the array Z. LDZ >= 1. * If COMPZ='V' or 'I', then LDZ >= N. * * WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) * On exit, if INFO >= 0, WORK(1) returns the optimal LWORK. * * LWORK (input) INTEGER * The dimension of the array WORK. LWORK >= max(1,N). * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * = 1,...,N: the QZ iteration did not converge. (A,B) is not * in Schur form, but ALPHAR(i), ALPHAI(i), and * BETA(i), i=INFO+1,...,N should be correct. * = N+1,...,2*N: the shift calculation failed. (A,B) is not * in Schur form, but ALPHAR(i), ALPHAI(i), and * BETA(i), i=INFO-N+1,...,N should be correct. * > 2*N: various "impossible" errors. * * Further Details * =============== * * Iteration counters: * * JITER -- counts iterations. * IITER -- counts iterations run since ILAST was last * changed. This is therefore reset only when a 1-by-1 or * 2-by-2 block deflates off the bottom. * * ===================================================================== * * .. Parameters .. * $ SAFETY = 1.0E+0 )
DHGEQZ()
DHGEQZ(String, String, String, int, int, int, double[][], double[][], double[], double[], double[], double[][], double[][], double[], int, intW)
DHGEQZ
public DHGEQZ()
DHGEQZ
public static void DHGEQZ(String job,
String compq,
String compz,
int n,
int ilo,
int ihi,
double a[][],
double b[][],
double alphar[],
double alphai[],
double beta[],
double q[][],
double z[][],
double work[],
int lwork,
intW info)
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