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数值方法
12
engines
Bisection method · halving the bracket
Bisection method · halving the bracket
区间[a,b]且f(a)f(b)<0 · 逐次折半 · 误差每步减半
f(a)·f(b) < 0 guarantees a root inside [a,b]; keep the half that still changes sign
Secant method · root finding without derivatives
Secant method · root finding without derivatives
x_{n+1}=x_n−f(x_n)(x_n−x_{n-1})/Δf · 无需导数 · 超线性
xₙ₊₁ = xₙ − f(xₙ)·(xₙ−xₙ₋₁)/(f(xₙ)−f(xₙ₋₁)): each secant line's x-intercept is the next iterate
Fixed-point iteration · cobweb diagram
Fixed-point iteration · cobweb diagram
x=g(x) · x_{n+1}=g(x_n) · |g'|<1收敛 · 蛛网图
Rewrite f(x)=0 as x=g(x), iterate xₙ₊₁=g(xₙ); converges iff |g'(x*)| < 1
Trapezoidal rule — area by trapezoids
Trapezoidal rule — area by trapezoids
∫≈h[½f0+f1+…+½fn] · 梯形逼近 · 误差O(h²)
∫f ≈ h[½f₀ + f₁ + … + ½fₙ]; the error shrinks like O(h²)
Simpson's rule — parabolic arcs
Simpson's rule — parabolic arcs
∫≈h/3[f0+4f1+2f2+…] · 抛物线 · O(h⁴) · 三次精确
∫f ≈ (h/3)[f₀ + 4f₁ + 2f₂ + … + fₙ]; error O(h⁴), exact for cubics
Finite differences — numerical derivative
Finite differences — numerical derivative
前差(f(x+h)−f(x))/h · 中心差/2h · 误差vs h
forward (f(x+h)−f(x))/h is O(h); central (f(x+h)−f(x−h))/(2h) is O(h²)
Euler's method · first-order ODE stepping
Euler's method · first-order ODE stepping
y_{n+1}=y_n+h·f · 一阶 · O(h) · 沿切线漂移
yₙ₊₁ = yₙ + h·f(xₙ,yₙ): follow the tangent slope one step at a time
Runge–Kutta 4 · high-accuracy ODE stepping
Runge–Kutta 4 · high-accuracy ODE stepping
RK4四斜率加权 · O(h⁴) · 远优于欧拉
yₙ₊₁ = yₙ + (h/6)(k₁+2k₂+2k₃+k₄): four slope samples per step
Lagrange interpolation · the polynomial through the points
Lagrange interpolation · the polynomial through the points
过n点唯一n−1次多项式 · 基Li(x) · Runge振荡
P(x) = Σ yᵢ·Lᵢ(x): one degree ≤ n−1 polynomial hitting every node
Gaussian elimination · solving A·x = b
Gaussian elimination · solving A·x = b
Ax=b · 前向消元化上三角+回代 · 主元/乘子
Edit the system, then step through forward elimination and back-substitution — exact fractions
LU decomposition · A = L·U, then solve A·x = b
LU decomposition · A = L·U, then solve A·x = b
A=L·U · 前代Ly=b后代Ux=y · 复用L/U
Factor A once into L and U; reuse it to solve for any right-hand side b in two triangular passes
Gauss–Seidel iteration · solving A·x = b
Gauss–Seidel iteration · solving A·x = b
立即用更新值 · 对角占优收敛 · 残差收缩
Step a diagonally-dominant system toward its solution — updated values reused immediately