Package jline.lib.qmam

Class Q_Sylvest

java.lang.Object

jline.lib.qmam.Q_Sylvest

public final class Q_Sylvest
extends Object

Method Summary

Modifier and Type	Method	Description
static ComplexMatrix	kronEye(ComplexMatrix A, int m)	Computes kron(A, eye(m)) for a complex matrix A.
static Matrix	qSylvest(ComplexMatrix U, ComplexMatrix T, Matrix B)	Solves the equation X*kron(A,I)+B*X=-I, where kron(A,I)=U*T*U' is given by its complex Schur decomposition (U unitary, T upper triangular).
static Matrix	qSylvest(ComplexMatrix U, ComplexMatrix T, Matrix V, Matrix NBAR, Matrix C)	Solves X*kron(A,I)+B*X=-C, with B given by its Hessenberg form V'*B*V=NBAR and kron(A,I)=U*T*U'.
static Map<String,ComplexMatrix>	schurDecomposition(Matrix A)	Computes the complex Schur decomposition of A, equivalent to MATLAB's [U,T] = schur(A,'complex') with A = U*T*U'.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Details

qSylvest

public static Matrix qSylvest(ComplexMatrix U,
 ComplexMatrix T,
 Matrix B)

Solves the equation X*kron(A,I)+B*X=-I, where kron(A,I)=U*T*U' is given by its complex Schur decomposition (U unitary, T upper triangular).

Mirrors MATLAB Q_Sylvest.m: B is reduced to upper Hessenberg form NBAR=V'*B*V, the transformed equation Y*T+NBAR*Y=F with F=-V'*U is solved column by column exploiting the triangularity of T, and the solution is mapped back as X=real(V*Y*U').

Parameters:

U - unitary factor of the complex Schur decomposition of kron(A,I)

T - upper triangular factor of the complex Schur decomposition of kron(A,I)

B - the coefficient matrix multiplying X from the left

Returns:

the real solution X

qSylvest

public static Matrix qSylvest(ComplexMatrix U,
 ComplexMatrix T,
 Matrix V,
 Matrix NBAR,
 Matrix C)

Solves X*kron(A,I)+B*X=-C, with B given by its Hessenberg form V'*B*V=NBAR and kron(A,I)=U*T*U'.

This is the right-hand-side form inlined by MATLAB Q_CT_MMAPK_PHK_1.m in its queue length recursions (F=-V'*Cn*U); with C=I it reduces to qSylvest(ComplexMatrix, ComplexMatrix, Matrix).

Parameters:

U - unitary factor of the complex Schur decomposition of kron(A,I)

T - upper triangular factor of the complex Schur decomposition of kron(A,I)

V - orthogonal factor of the Hessenberg decomposition of B

NBAR - upper Hessenberg form V'*B*V of B

C - the right-hand side

Returns:

the real solution X

schurDecomposition

public static Map<String,ComplexMatrix> schurDecomposition(Matrix A)

Computes the complex Schur decomposition of A, equivalent to MATLAB's [U,T] = schur(A,'complex') with A = U*T*U'.

Parameters:

A - the matrix to decompose

Returns:

a map with keys "U" (unitary) and "T" (upper triangular)

kronEye

public static ComplexMatrix kronEye(ComplexMatrix A,
 int m)

Computes kron(A, eye(m)) for a complex matrix A.

Parameters:

A - the complex matrix

m - the order of the identity factor

Returns:

the Kronecker product kron(A, eye(m))