縮基法(RBM) 是對參數化的曲線求逼近解的一個方法，基本上乃使用投影法將解曲線投射到解空間的一子空間中，如此一來，可將原問題轉換成一較小的系統，並經由數值計算出小系統的解，來求得大系統的一逼近解。在本篇論文中主要的乃探討RBM在常微分方程組初始值問題上的應用，並發展一套含有誤差控制的演算法。

本篇論文中所採用的ODE Solver 乃由Gordon 和Shampine 基於Adams PECE方法所發展的。在求解的過程中，對於計算解誤差的控制我們除了利用ODE Solver 的誤差估計，另外我們又發展對縮基解(reduced basis solution) 的後(Aposteriori)

誤差估計，以確保數值計算解的準確性。我們所考慮使用的子空間有三種Taylor, Lagrange , Hermite 。同時為了要增加數值的特定性及簡化小系統的求解工作，我們先行將子空間的基底直交化。因此，除了誤差的控制外，我們也討論了roundoff error 對向量直交化及形成小系統時所造成的影響，並設立誤差標準以判別何時誤差過大到嚴重影響縮基解的準確度。

本篇論文的目的是希望利用RBM發展出一套解常微分方程組初始值問題的求算法，以期計算解能在較短的時間內準確的被計算出來。

The reduced basis method(RBM) is a scheme for approximating parametric solution curves. The basic technique of RBM is projection. By applying the method, we can find an approximate solution of the original system which satisfies a system of smaller size. In this paper, we mainly concern the applications of RBM for ODE initial value problems and develop an algorithm which contains a set of error controls.

The ODE solver used in this paper is developed by Gordon and Shampine based on Adams PECE formulas. To assure the accuracy of the reduced basis approximation, we set up an appropriate automatic error control in calling GS solver and develop an a posteriori error estimate to keep the reduction error under control.

The subspaces considered are Taylor, Lagrange and Hermite subspaces.In the meantime, in order to improve the numerical stability and simplify the computation of the reduced basis solution, we orthogonalize the generators of reduced subspaces. We also discuss the roundoff errors in the orthogonalization process and build up a criterion for identifying the case the accuracy of the reduced basis solution up a criterion for identifying the case the accuracy of the reduced basis solution is destroyed by the errors.

The aim of this paper is to develop an algorithm to solve the ODE initial value problems efficiently.

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