Control theory¶
The following algorithms draw heavily from the second chapter of Nicholas J. Higham’s “Functions of Matrices: Theory and Computation” and depend heavily on the matrix sign function. They have only undergone cursory testing.
Sylvester¶
As long as both \(A\) and \(B\) only have eigenvalues in the open right-half plane, the following routines solve for \(X\) in the Sylvester equation
\[A X + X B = C\]
via computing \(\text{sgn}(W)\), where
\[\begin{split}W = \begin{pmatrix} A & -C \\ 0 & -B \end{pmatrix}.\end{split}\]
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void
Sylvester(const Matrix<F> &A, const Matrix<F> &B, const Matrix<F> &C, Matrix<F> &X, SignCtrl<Base<F>> signCtrl = SignCtrl<Base<F>>())¶
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void
Sylvester(const DistMatrix<F> &A, const DistMatrix<F> &B, const DistMatrix<F> &C, DistMatrix<F> &X, SignCtrl<Base<F>> signCtrl = SignCtrl<Base<F>>())¶
One may also directly pass in \(W\) in order to save space.
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void
Sylvester(int m, Matrix<F> &W, Matrix<F> &X, SignCtrl<Base<F>> signCtrl = SignCtrl<Base<F>>())¶
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void
Sylvester(int m, DistMatrix<F> &W, DistMatrix<F> &X, SignCtrl<Base<F>> signCtrl = SignCtrl<Base<F>>())¶
Lyapunov¶
A special case of the Sylvester solver, where \(B = A^H\).
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void
Lyapunov(const Matrix<F> &A, const Matrix<F> &C, Matrix<F> &X, SignCtrl<Base<F>> signCtrl = SignCtrl<Base<F>>())¶
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void
Lyapunov(const DistMatrix<F> &A, const DistMatrix<F> &C, DistMatrix<F> &X, SignCtrl<Base<F>> signCtrl = SignCtrl<Base<F>>())¶
Algebraic Ricatti¶
Under suitable conditions, the following routines solve for \(X\) in the algebraic Ricatti equation
\[X K X - A^H X - X A = L,\]
where both \(K\) and \(L\) are Hermitian.
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void
Ricatti(UpperOrLower uplo, const Matrix<F> &A, const Matrix<F> &K, const Matrix<F> &L, Matrix<F> &X, SignCtrl<Base<F>> signCtrl = SignCtrl<Base<F>>())¶
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void
Ricatti(UpperOrLower uplo, const DistMatrix<F> &A, const DistMatrix<F> &K, const DistMatrix<F> &L, DistMatrix<F> &X, SignCtrl<Base<F>> signCtrl = SignCtrl<Base<F>>())¶
Alternatively, one may directly fill the matrix \(W\).
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void
Ricatti(DistMatrix<F> &W, DistMatrix<F> &X, SignCtrl<Base<F>> signCtrl = SignCtrl<Base<F>>())¶