科学计算复习

Scientific Computing Review Outline

Floating-Point Number System

Problem 1.3

Let $x_f$ and $y_f$ be adjacent floating-point numbers. You can assume they are positive and normal floats.
What is the minimum possible distance between $x_f$ and $y_f$?Maximum possible distance?

Solving A Nonlinear Equation

Key points: Newton's Method,Secant Method. Bisection Method is simple.

Problem 2.24

This problem considers solving $f(x)=0$,where $$f(x)=x^3-3x-1.$$
1.Using Newton's method, take $x_0 =2$ and calculate $x_1$.After this,calculate $x_2$.
2.Using Secant method, take $x_0 =2,x_1=1.9$ and calculate $x_1$.

Others:Order of Convergence

Matrix Equations

Key points:Doolittle factorization,Crout factorization,Cholesky factorization

Problem 3.5

Consider the matrix

\[A=\begin{pmatrix} 3 & -1& -1 \\ -1 & 3&- 2\\ -1&-2 &5 \end{pmatrix}\]
1. Find $||A||_{1},A^{-1}, \kappa_{\infty}(A)$.
2. Find the Doolittle factorization, Crout factorization of A.
3. Using a Doolittle factorization, solve $AX=b,b=(-1,1,-4)^{T}$.
4. Find the Cholesky factorization.

Eigenvalue Problems

Key points: Power Method,QR Factorization,SVD

Problem 4.3

The symmetric matrix

\[A=\begin{pmatrix} 11 & 7& -4\\ 7 & 11&4\\ -4&4 &-10 \end{pmatrix}\]
has eigenvectors $x_1 =(1,1,0)^{T}, x_2 =(1,0,2)^{T},x_3 =(0,−2,1)^{T}$.
1. What are the corresponding eigenvalues?
2. Assuming the starting vector $y_0=(1,1,1)^{T}$,what eigenvalue will the power method converge to and what will be the resulting eigenvector?
3. Assuming $C_0=A$,find $C_1$ when using the QR method.
4. When using the QR method, what matrix does the $C_k$ matrices converge to?

Problem SVD

\[A=\begin{pmatrix} 1 & 2\\ 0 & 1\\ 1&0 \end{pmatrix}\]
1. Find a SVD for $A$.
2. Find a SVD for $-3A$.
3. Find $||A||_2$.

Interpolation

Key points: Lagrange,Cubic Spline
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Problem 5.1

In this problem the data are: $(x_1,y_1)=(0,0), (x_2,y_2)=(1,1)$,and $(x_3,y_3)=(2,3)$.
1.Find the global interpolation polynomial that fits these data.
(Do not use method of undetermined coefficients.)
2.Find the natural cubic spline that fits these data.

Numerical Integration

Key points:Midpoint, Trapezoid, Simpson,Hermite
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Problem 6.2

This problem concerns using numerical methods to calculate the integral

\[I = \int_{-1}^{1} e^{-2x} \,\mathrm{d}x. \]
1. Using the composite trapezoidal rule, and 4 subintervals, find an approximate value for the integral. What is the error?
2. Using the composite Simpson's rule, and 4 subintervals, find an approximate value for the integral. What is the error?
3. Using the composite Hermite rule, and 4 subintervals, find an approximate value for the integral. What is the error?
4. Using the composite trapezoidal rule, how small does the step size $h$ have to be to guarantee that the numerical error is less than $10^{-6}$ ?
5. Using the composite Simpson's rule, how small does the step size $h$ have to be to guarantee that the numerical error is less than $10^{-6}$?

Numerical differentiation