Constructing Riccati differential equations with known analytic solutions for numerical experiments

Choi, Chiu; Laub, Alan J.

doi:10.1109/9.52297

Cited by 12 publications

(6 citation statements)

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“…This theorem was proved for the maximal solution of (2.54) (see Section 2.17) by Molinari [59], and was extended to the class of admissible solutions in [60]. We remark that, given an admissible solution X, the fact that F(Z) has a factorization of the form (2.57) (with D = R + B* X B) is just a computation.…”

Section: The Discrete Riccati Equation: the Frequency Domain Approachmentioning

confidence: 87%

“…As solutions for which R+ B* X B > 0 are of particular interest, the following result gives more information on this case (see [60] If the conditions (a), (b), (c) (or one of them) of Theorem 2.37 are valid, then there is a maximal admissible solution X+ of (2.55), i.e., such that X+ ~ X for any other admissible solution X. As solutions for which R+ B* X B > 0 are of particular interest, the following result gives more information on this case (see [60] If the conditions (a), (b), (c) (or one of them) of Theorem 2.37 are valid, then there is a maximal admissible solution X+ of (2.55), i.e., such that X+ ~ X for any other admissible solution X.…”

Section: The Discrete Riccati Equation: the Frequency Domain Approachmentioning

confidence: 99%

“…Using some detailed infonnation on factorization of rational matrix functions which are self-adjoint on the unit circle, the following theorem is proved in reference [60]. There is a marked contrast between this proof and that of the analogous Theorem 2.6.…”

Section: Geometric Theory For the Discrete Riccati Equationmentioning

confidence: 99%

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The Riccati Equation

1991

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Section: The Discrete Riccati Equation: the Frequency Domain Approachmentioning

confidence: 87%

Section: The Discrete Riccati Equation: the Frequency Domain Approachmentioning

confidence: 99%

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The Riccati Equation

1991

213

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“…Example 2. This is taken from [10], where a few time-invariant cases with known solutions are examined. The MRDE we tested is…”

Section: Example 1 ([23 34])mentioning

confidence: 99%

GIP integrators for Matrix Riccati Differential Equations

Garrett¹,

Li²

2014

Applied Mathematics and Computation

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“…It is taken from [9], where a few time-invariant cases with known solutions are examined. The MRDE we tested is…”

Section: Examplementioning

confidence: 99%

A family of Anadromic numerical methods for matrix Riccati differential equations

Li¹,

Kahan²

2012

Math. Comp.

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Abstract. Matrix Riccati Differential Equations (MRDEs)X = A 21 − XA 11 + A 22 X − XA 12 X, X(0) = X 0 , where A ij ≡ A ij (t), appear frequently throughout applied mathematics, science, and engineering. Naturally, the existing conventional Runge-Kutta methods and linear multi-step methods can be adapted to solve MRDEs numerically. Indeed, they have been adapted. There are a few unconventional numerical methods, too, but they are suited more for time-invariant MRDEs than time-varying ones. For stiff MRDEs, existing implicit methods which are preferred to explicit ones require solving nonlinear systems of equations (of possibly much higher dimensions than the original problem itself of, for example, implicit Runge-Kutta methods), and thus they can pose implementation difficulties and also be expensive.In the past, the property of an MRDE which has been most preserved is the symmetry property for a symmetric MRDE; many other crucial properties have been discarded. Besides the symmetry property, our proposed methods also preserve two other important properties -Bilinear Rational Dependence on the initial value, and a Generalized Inverse Relation between an MRDE and its complementary MRDE. By preserving the generalized inverse relation, our methods are accurately able to integrate an MRDE whose solution has singularities. By preserving the property of bilinear dependence on the initial value, our methods also conserve the rank of change to the initial value and a solution's monotonicity property.Our methods are anadromic, 1 meaning if an MRDE is integrated by one of our methods from t=τ to τ + θ and then integrated backward from t=τ + θ to τ using the same method, the value at t=τ is recovered in the absence of rounding errors. This implies that our methods are necessarily of even order of convergence. For time-invariant MRDEs, methods of any even order of convergence are established, while for time-varying MRDEs, methods of order as high as 10 are established; but only methods of order up to 6 are stated in detail.Our methods are semi-implicit, in the sense that there are no nonlinear systems of matrix equations to solve, only linear ones, unlike any pre-existing implicit method. Given the availability of high quality codes for linear matrix equations, our methods can easily be implemented and embedded into any application software package that needs a robust MRDE solver.Numerical examples are presented to support our claims.

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Constructing Riccati differential equations with known analytic solutions for numerical experiments

Cited by 12 publications

References 0 publications

The Riccati Equation

The Riccati Equation

GIP integrators for Matrix Riccati Differential Equations

A family of Anadromic numerical methods for matrix Riccati differential equations

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