By P.J.Van Der Houwen

ISBN-10: 0720423503

ISBN-13: 9780720423501

ISBN-10: 0720423724

ISBN-13: 9780720423723

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**Example text**

J / M /'(Λ-,,+ Γ, αι, α2, · · . 23) as simple as possible. 23) then are linear and therefore easily solved. However, also rational functions / give rise to linear equations for the a·. For future reference some polynomial and rational interpolation formulas are given below. 24) r = A and rational expressions of vectors should be interpreted as componentwise operations on these vectors. 28) yn+v h„y'„]v*; = y„ + h„yliv+[3(yn + i--yn)-hn(2y'„+y'„+i)]v1 + [2(yn-yn+i) + h„(y'„+y'n+1)]v:i; = y„ + h„y'nv+ \h2nynv2 + [4(j>„+ i-yn)-hn(3y'n+y'n+ i)- A ^ V + [Xyn~yn+\) + hf{2y'ni-y'n+i) + -hti2„y,n]v/i.

The first generalized Runge-Kutta method using only one evaluation of the Jacobian matrix per integration step has been given by ROSENBROCK [1963]. (ζ). \ , l-0-lV2)z 46 SINGLE-STEP METHODS [CH. 2, § 3 Note that two evaluations of the right hand side/and one LU-decomposition are involved. CALAHAN [1968] pointed out that it is possible to construct a third order method which is also based on two evaluations o f / a n d one LU-decomposition. His method is generated by the coefficient functions (235) 4(1-(UJV3)Z) .

And j n + 1 , y'n+19 j / / + 1 , . . We now seek an approximation toy at Λ: = xn+ h with 0 < h < hn. A formula which provides this approximation is called an interpolation formula. A general approach for constructing interpolation formulas reads as follows: firstly, we select a function / = I(x; a i9 a2, . . , am) such that it gives a reasonable representation of y when the parameter vectors a. are suitably 26 INTRODUCTION [ C H . 1, § 6 chosen; the next step is to solve these parameters from the equations /(*„; αι, α 2 , .

### Construction of Integration Formulas for Initial Value Problems by P.J.Van Der Houwen

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