org.netlib.lapack
Class DTGEVC
java.lang.Object
org.netlib.lapack.DTGEVC
public class DTGEVC
 extends java.lang.Object
DTGEVC is a simplified interface to the JLAPACK routine dtgevc.
This interface converts Javastyle 2D rowmajor arrays into
the 1D columnmajor 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
* =======
*
* DTGEVC computes some or all of the right and/or left generalized
* eigenvectors of a pair of real upper triangular matrices (A,B).
*
* The right generalized eigenvector x and the left generalized
* eigenvector y of (A,B) corresponding to a generalized eigenvalue
* w are defined by:
*
* (A  wB) * x = 0 and y**H * (A  wB) = 0
*
* where y**H denotes the conjugate tranpose of y.
*
* If an eigenvalue w is determined by zero diagonal elements of both A
* and B, a unit vector is returned as the corresponding eigenvector.
*
* If all eigenvectors are requested, the routine may either return
* the matrices X and/or Y of right or left eigenvectors of (A,B), or
* the products Z*X and/or Q*Y, where Z and Q are input orthogonal
* matrices. If (A,B) was obtained from the generalized realSchur
* factorization of an original pair of matrices
* (A0,B0) = (Q*A*Z**H,Q*B*Z**H),
* then Z*X and Q*Y are the matrices of right or left eigenvectors of
* A.
*
* A must be block upper triangular, with 1by1 and 2by2 diagonal
* blocks. Corresponding to each 2by2 diagonal block is a complex
* conjugate pair of eigenvalues and eigenvectors; only one
* eigenvector of the pair is computed, namely the one corresponding
* to the eigenvalue with positive imaginary part.
*
* Arguments
* =========
*
* SIDE (input) CHARACTER*1
* = 'R': compute right eigenvectors only;
* = 'L': compute left eigenvectors only;
* = 'B': compute both right and left eigenvectors.
*
* HOWMNY (input) CHARACTER*1
* = 'A': compute all right and/or left eigenvectors;
* = 'B': compute all right and/or left eigenvectors, and
* backtransform them using the input matrices supplied
* in VR and/or VL;
* = 'S': compute selected right and/or left eigenvectors,
* specified by the logical array SELECT.
*
* SELECT (input) LOGICAL array, dimension (N)
* If HOWMNY='S', SELECT specifies the eigenvectors to be
* computed.
* If HOWMNY='A' or 'B', SELECT is not referenced.
* To select the real eigenvector corresponding to the real
* eigenvalue w(j), SELECT(j) must be set to .TRUE. To select
* the complex eigenvector corresponding to a complex conjugate
* pair w(j) and w(j+1), either SELECT(j) or SELECT(j+1) must
* be set to .TRUE..
*
* N (input) INTEGER
* The order of the matrices A and B. N >= 0.
*
* A (input) DOUBLE PRECISION array, dimension (LDA,N)
* The upper quasitriangular matrix A.
*
* LDA (input) INTEGER
* The leading dimension of array A. LDA >= max(1, N).
*
* B (input) DOUBLE PRECISION array, dimension (LDB,N)
* The upper triangular matrix B. If A has a 2by2 diagonal
* block, then the corresponding 2by2 block of B must be
* diagonal with positive elements.
*
* LDB (input) INTEGER
* The leading dimension of array B. LDB >= max(1,N).
*
* VL (input/output) DOUBLE PRECISION array, dimension (LDVL,MM)
* On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
* contain an NbyN matrix Q (usually the orthogonal matrix Q
* of left Schur vectors returned by DHGEQZ).
* On exit, if SIDE = 'L' or 'B', VL contains:
* if HOWMNY = 'A', the matrix Y of left eigenvectors of (A,B);
* if HOWMNY = 'B', the matrix Q*Y;
* if HOWMNY = 'S', the left eigenvectors of (A,B) specified by
* SELECT, stored consecutively in the columns of
* VL, in the same order as their eigenvalues.
* If SIDE = 'R', VL is not referenced.
*
* A complex eigenvector corresponding to a complex eigenvalue
* is stored in two consecutive columns, the first holding the
* real part, and the second the imaginary part.
*
* LDVL (input) INTEGER
* The leading dimension of array VL.
* LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise.
*
* VR (input/output) DOUBLE PRECISION array, dimension (LDVR,MM)
* On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
* contain an NbyN matrix Q (usually the orthogonal matrix Z
* of right Schur vectors returned by DHGEQZ).
* On exit, if SIDE = 'R' or 'B', VR contains:
* if HOWMNY = 'A', the matrix X of right eigenvectors of (A,B);
* if HOWMNY = 'B', the matrix Z*X;
* if HOWMNY = 'S', the right eigenvectors of (A,B) specified by
* SELECT, stored consecutively in the columns of
* VR, in the same order as their eigenvalues.
* If SIDE = 'L', VR is not referenced.
*
* A complex eigenvector corresponding to a complex eigenvalue
* is stored in two consecutive columns, the first holding the
* real part and the second the imaginary part.
*
* LDVR (input) INTEGER
* The leading dimension of the array VR.
* LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise.
*
* MM (input) INTEGER
* The number of columns in the arrays VL and/or VR. MM >= M.
*
* M (output) INTEGER
* The number of columns in the arrays VL and/or VR actually
* used to store the eigenvectors. If HOWMNY = 'A' or 'B', M
* is set to N. Each selected real eigenvector occupies one
* column and each selected complex eigenvector occupies two
* columns.
*
* WORK (workspace) DOUBLE PRECISION array, dimension (6*N)
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = i, the ith argument had an illegal value.
* > 0: the 2by2 block (INFO:INFO+1) does not have a complex
* eigenvalue.
*
* Further Details
* ===============
*
* Allocation of workspace:
*   
*
* WORK( j ) = 1norm of jth column of A, above the diagonal
* WORK( N+j ) = 1norm of jth column of B, above the diagonal
* WORK( 2*N+1:3*N ) = real part of eigenvector
* WORK( 3*N+1:4*N ) = imaginary part of eigenvector
* WORK( 4*N+1:5*N ) = real part of backtransformed eigenvector
* WORK( 5*N+1:6*N ) = imaginary part of backtransformed eigenvector
*
* Rowwise vs. columnwise solution methods:
*     
*
* Finding a generalized eigenvector consists basically of solving the
* singular triangular system
*
* (A  w B) x = 0 (for right) or: (A  w B)**H y = 0 (for left)
*
* Consider finding the ith right eigenvector (assume all eigenvalues
* are real). The equation to be solved is:
* n i
* 0 = sum C(j,k) v(k) = sum C(j,k) v(k) for j = i,. . .,1
* k=j k=j
*
* where C = (A  w B) (The components v(i+1:n) are 0.)
*
* The "rowwise" method is:
*
* (1) v(i) := 1
* for j = i1,. . .,1:
* i
* (2) compute s =  sum C(j,k) v(k) and
* k=j+1
*
* (3) v(j) := s / C(j,j)
*
* Step 2 is sometimes called the "dot product" step, since it is an
* inner product between the jth row and the portion of the eigenvector
* that has been computed so far.
*
* The "columnwise" method consists basically in doing the sums
* for all the rows in parallel. As each v(j) is computed, the
* contribution of v(j) times the jth column of C is added to the
* partial sums. Since FORTRAN arrays are stored columnwise, this has
* the advantage that at each step, the elements of C that are accessed
* are adjacent to one another, whereas with the rowwise method, the
* elements accessed at a step are spaced LDA (and LDB) words apart.
*
* When finding left eigenvectors, the matrix in question is the
* transpose of the one in storage, so the rowwise method then
* actually accesses columns of A and B at each step, and so is the
* preferred method.
*
* =====================================================================
*
* .. Parameters ..
Method Summary 
static void 
DTGEVC(java.lang.String side,
java.lang.String howmny,
boolean[] select,
int n,
double[][] a,
double[][] b,
double[][] vl,
double[][] vr,
int mm,
intW m,
double[] work,
intW info)

Methods inherited from class java.lang.Object 
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait 
DTGEVC
public DTGEVC()
DTGEVC
public static void DTGEVC(java.lang.String side,
java.lang.String howmny,
boolean[] select,
int n,
double[][] a,
double[][] b,
double[][] vl,
double[][] vr,
int mm,
intW m,
double[] work,
intW info)