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{\bf MATH 269A.  HW \#4} (due on Monday, November 5)

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{\large 
{\bf 1.} 

(a) Determine the coefficients in 
$$y_{i+1}=a_0 y_i +h\Big\{b_{1}f(x_{i+1},y_{i+1})+b_0 f(x_i,y_i)\Big\}$$
if the formula is to be exact for polynomials of degree 2 (consider polynomials
$x^j$, for $j=0,\ 1,\ 2$). 

(b) Using the coefficients found in (a), find the order of the method and a
simple expression of the local truncation error
(use Taylor's expansion and express the local
truncation error function of a higher derivative of the exact solution). 

(c) What equivalent definition for the order of the method can be formulated ?

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{\bf 2.} Use the quadratic interpolant to $f(x,y(x))$ at $x_{i}$, 
$x_{i-1}$, $x_{i-2}$ to obtain the numerical method 
$$y_{i+1}=y_{i-3}+\frac{4h}{3}\Big\{2f(x_i,y_i)-f(x_{i-1},y_{i-1})+2f(x_{i-2},
y_{i-2})\Big\}.$$
What is the order of the method given by the local truncation error ? 
  
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{\bf 3.} For $q=1$ determine the coefficients $\beta_{qi}$ in 
Nystr\"om's formula
$$y_{p+1}=y_{p-1}+h\Big\{\beta_{10}f(x_p,y_p)+\beta_{11}f(x_{p-1},y_{p-1})\Big\}.$$

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{\bf 4.} Implement the fourth order Adams predictor-corrector method:
\begin{eqnarray*}
y_{i+1}^{(0)}=y_i+\frac{h}{24}\Big\{55 f(x_i,y_i)-59f(x_{i-1},y_{i-1})+
37f(x_{i-2},y_{i-2})-9f(x_{i-3},y_{i-3})\Big\}\\
y_{i+1}=y_i+\frac{h}{24}\Big\{9f(x_{i+1}, y_{i+1}^{(0)})+19f(x_i,y_i)-5f(x_{i-1},y_{i-1})+f(x_{i-2},y_{i-2})\Big\}
\end{eqnarray*}

Verify the correctness of your implementation by applying it to the problem 
$$y'(x)=\frac{3x^2+4x+2}{2(y-1)}, \ \ y(0)=-1.$$

(a) Explain how you obtained your starting values.

(b) Give the results at $x=2$ of your algorithm for stepsizes $h=0.1$, $0.05$,
$0.025$, and $0.0125$, and compare them with the exact solution at $x=2$.  

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