baixiaohub / ninumerical

Numerical method samples in python & pyplot.

NiNumerical

---

Numerical method samples in python & pyplot. Refer to <<Numerical Methods for Engineers 7th>>.

pip install -r requirements.txt

to install dependencies.

---

• RootFinding
  • Bracketing (simple)
    • Bisection Method
    • False Position Method
  • Open (faster, but may diverge or cycle)
    • Fixed Point Iteration
    • Newton-Raphson Method (needs f and f') $ x_i+1 = x_i + f(x_i)/f'(x_i) $
      • Important Points: Newton’s method is useful in cases of large values of f’(x) i.e. when the graph of f(x) while crossing the x-axis is nearly vertical. May converge slowly.
      • It is not preferred when the graph of f(x) is nearly horizontal where it crosses the x-axis as the values of f’(x) have negative values in this case.
      • It is sensitive to starting value.
      • Convergence fails if the starting point is nor near the root.
      • The formula converges provided the initial approximation x0 is chosen sufficiently close to the root.
      • It is generally used to improve the result obtained by the other methods.
      • It has quadratic convergence i.e. order of convergence is 2. The subsequent error at each step is proportional to the square of the error at the previous step.
    • Modified Newton-Raphson Method $ x_i+1 = x_i + f(x_i)f'(x_i) / ( [f'(x_i)]^2 -f(x_i)f''(x_i) ) $
    • Secant Method (needs 2 points) $ x_i+1 = x_i + u(x_i)/u(x_i) $
    • Modified Secant Method $ x_i+1 = x_i - u(x_i)(x_i-1 + x_i)/(u(x_i-1) - u(x_i)) $
    • Inverse Quadratic Interpolation (needs 3 points)
  • Graphical (for debug)
• LinearAlgebra
• CurveFitting
• Differentiation
• Integration
• Optimization
  • Convex
  • Nonlinear
  • more
• ODE
• PDE

Sources of Error

• Modeling Error
  • Formulation Error
  • Data Uncertainty
  • *Blunders
• Numerical Error
  • Truncation Error
    • Taylor Series
  • Roundoff Error
    • Quantization Error
    • Numerical Manipulations
      • Adding large and small numbers
      • Subtractive cancellation