Conditioning of Collocation Matrices and Discrete Green’s Functions
Copyright policy )Conditioning of Collocation Matrices and Discrete Green’s Functions
The boundary value problem \(Lu(x)=u^{(m)}(x)-\sum^{m}_{i=1}C_ i(x)u^{(i-1)}(x)=f(x)\), \(x\in [a,b]\); \(B_ aZ(u(a))+B_ b(u(b))=\beta,\) \(B_ a,B_ b\in R^{m\times m}\), \(Z(u(x))=(u(x),u'(x),...,u^{(m-1)}(x))^ T\) is considered. The stability and related conditioning of spline collocation matrices is analysed. Hermite-type and B-spline bases are used. The corresponding matrix \(A_ c\) has an obvious similarity to that obtained in applying multiple shooting. By examining this relationship the explicit form for \(A_ c^{-1}\) and estimate \(K_ 0\tau (\Delta)\leq K(A_ c)\leq K_ 1\tau (\Delta)\) are obtained. The constants \(K_ 0\) and \(K_ 1\) are independent of the grid \(\Delta\), \(\tau\) (\(\Delta)\) is expressed via the Green function and the fundamental solution matrix of the corresponding first order system.
Numerical solution of boundary value problems involving ordinary differential equations, conditioning, Numerical computation of matrix norms, conditioning, scaling, Linear boundary value problems for ordinary differential equations, spline collocation, multiple shooting, stability, discrete Green's function, Stability and convergence of numerical methods for ordinary differential equations
Numerical solution of boundary value problems involving ordinary differential equations, conditioning, Numerical computation of matrix norms, conditioning, scaling, Linear boundary value problems for ordinary differential equations, spline collocation, multiple shooting, stability, discrete Green's function, Stability and convergence of numerical methods for ordinary differential equations
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