org.netlib.lapack
Class Dtgevc
java.lang.Object
org.netlib.lapack.Dtgevc
public class Dtgevc
- extends java.lang.Object
Following is the description from the original
Fortran source. For each array argument, the Java
version will include an integer offset parameter, so
the arguments may not match the description exactly.
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 real-Schur
* 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 1-by-1 and 2-by-2 diagonal
* blocks. Corresponding to each 2-by-2 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 quasi-triangular 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 2-by-2 diagonal
* block, then the corresponding 2-by-2 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 N-by-N 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 N-by-N 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 i-th argument had an illegal value.
* > 0: the 2-by-2 block (INFO:INFO+1) does not have a complex
* eigenvalue.
*
* Further Details
* ===============
*
* Allocation of workspace:
* ---------- -- ---------
*
* WORK( j ) = 1-norm of j-th column of A, above the diagonal
* WORK( N+j ) = 1-norm of j-th 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 back-transformed eigenvector
* WORK( 5*N+1:6*N ) = imaginary part of back-transformed 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 i-th 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 = i-1,. . .,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 j-th 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 j-th 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 _select_offset,
int n,
double[] a,
int _a_offset,
int lda,
double[] b,
int _b_offset,
int ldb,
double[] vl,
int _vl_offset,
int ldvl,
double[] vr,
int _vr_offset,
int ldvr,
int mm,
intW m,
double[] work,
int _work_offset,
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 _select_offset,
int n,
double[] a,
int _a_offset,
int lda,
double[] b,
int _b_offset,
int ldb,
double[] vl,
int _vl_offset,
int ldvl,
double[] vr,
int _vr_offset,
int ldvr,
int mm,
intW m,
double[] work,
int _work_offset,
intW info)