科学计算复习

Scientific Computing Review Outline

Author: Hu Si
Date: 2025-11-16

Floating-Point Number System

Example Problem 1.3
Let ( x_f ) and ( y_f ) be adjacent floating-point numbers (positive normal floats).
What is the minimum/maximum possible distance between them?

Solving A Nonlinear Equation

Key Methods: Newton's Method, Secant Method, Bisection Method
Order of Convergence

Example Problem 2.24
Solve ( f(x) = x^3 - 3x - 1 = 0 )

1. Newton's Method ((x_0=2)):
   • ( x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} )
   • ( x_2 = \text{calculate similarly} )
2. Secant Method ((x_0=2, x_1=1.9)):
   • ( x_2 = x_1 - f(x_1)\frac{x_1 - x_0}{f(x_1) - f(x_0)} )

Matrix Equations

Key Factorizations: Doolittle, Crout, Cholesky

Example Problem 3.5
Matrix ( A = \begin{pmatrix} 3 & -1 & -1 \ -1 & 3 & -2 \ -1 & -2 & 5 \end{pmatrix} )

1. Compute ( |A|1 ), ( A^{-1} ), ( \kappa\infty(A) )
2. Find Doolittle/Crout factorizations
3. Solve ( A\mathbf{x} = \mathbf{b} ) with ( \mathbf{b} = (-1,1,-4)^T )
4. Cholesky factorization

Eigenvalue Problems

Key Methods: Power Method, QR Factorization, SVD

Example Problem 4.3
Symmetric matrix ( A = \begin{pmatrix} 11 & 7 & -4 \ 7 & 11 & 4 \ -4 & 4 & -10 \end{pmatrix} )
Eigenvectors: ( \mathbf{x}_1=(1,1,0)^T ), ( \mathbf{x}_2=(1,0,2)^T ), ( \mathbf{x}_3=(0,-2,1)^T )

1. Find corresponding eigenvalues
2. Power Method convergence with ( \mathbf{y}_0=(1,1,1)^T )
3. QR Method iteration (( C_0=A ))
4. Convergence target of ( C_k ) matrices

SVD Example
Matrix ( A = \begin{pmatrix} 1 & 2 \ 0 & 1 \ 1 & 0 \end{pmatrix} )

1. Compute SVD
2. SVD of ( -3A )
3. Compute ( |A|_2 )

Interpolation

Key Methods: Lagrange, Cubic Spline, Chebyshev

Example Problem 5.1
Data: ( (0,0), (1,1), (2,3) )

1. Global interpolation polynomial (avoid undetermined coefficients)
2. Natural cubic spline

Chebyshev Interpolation
Quadratic polynomial for ( f(x)=e^x ) on ([-1,1]) with error bound

Numerical Integration

Key Methods: Midpoint, Trapezoid, Simpson, Hermite

Example Problem 6.2
Integral ( I = \int_{-1}^{1} e^{-2x} dx )

1. Composite Trapezoidal (4 intervals): value & error
2. Composite Simpson (4 intervals): value & error
3. Composite Hermite (4 intervals): value & error
4. Trapezoidal step size for error < ( 10^{-6} )
5. Simpson step size for error < ( 10^{-6} )

Numerical Differentiation
Truncation error of:
[
f'(x_0) \approx \frac{1}{2h}[4f(x_0+h) - 3f(x_0) - f(x_0+2h)]
]

Initial Value Problems

Key Methods: Euler, Backward Euler, Classic Runge-Kutta