[Notes] Introduction to Computational Mathematics

Textbook: Numerical Optimization, J. Nocedal and S. J. Wright.

 

 Textbook: Numerical Computing with MATLAB, Mooler

與其說這些筆記是上面這兩本書的內容，不如說是薛克民老師自己整理的綱要，以下是我上他的「計算數學導論」 的筆記，我認為雖然這是數學系的必修，但是電機系會用到MATLAB的人都應該要學一學。

筆記中一些理論可能可以跳過，但是強烈推薦這門課的作業，有趣又有啟發性。

Notes Floating Point System Conditioning Number $Kp$     Forward/Backward Error Stability of an Algorithm and Efficient Digit Gauss Elimination     LU Decomposition         forward/backward substitution     with partial/complete pivoting     QR Decompsition Thm: Backward substitution is backward stable Cholesky Algorithm and Symmetric Positive-Definite Matrix Wilkinson Theorem,  Growth Factor and its Worst Case Root-Locating Methods of a Non-Linear Function     1-dim         Newton's Method         Secant Method         Bisection Method     $n$-dim         Newton's Method         Quasi-Newton/Broyden's Method/Secant Updating Theory of Convergence Fixed Point Iteration     Contraction Mapping Theorem Notes  Unconstrained Optimization     1-dim         Golden Section         Newton's Method     $n$-dim         Newton's Method         Quasi-Newton's Method(BFGS Method/Hession Updating)             Line Search             Sherman-Morrison-Woodbury Theorem         The Steapest Descent Method and Descent Direction             Special Case of Quadratic Form Interpolation     Vandermode Approach (Monomial Basis)     Lagrange Approach         Barycentric Formula     Newton Approach Convergence/Lebesguq Constant Equidistant Nodes/Chebyshev Nodes Runge Phenomenon Spline Function (Cubic Spline) Notes PCHIP     Not-A-Knot BC Numerical Differentiation     Lagrange Approach     Finite Difference         Method of Undetermined Coefficients     Collatz's Formula(Pade's Approximation) Numerical Intergration     Newton-Cotes Method     Clenshaw-Curtis Method     Gauss Quadrature         Gauss-Legendre Quadrature   Notes         Christoffle-Darboux Identity     Monte-Carlo Method Acceleration by Extrapolation     Aitken Method     Richardson Method     Romberg Integration     Euler-Maclaurin Formula Solving ODE by Numerical Method     Euler Method         Backward Euler Method     Trapezoidal Method     Midpoint Method (Leapfrog) Dahlquist Theorem Lipschitz Continuous Functions LMM(Linear Multistep Method)     Local Truncation Error     Characteristic Polynomial and Root Conditions     Absolutely Stable and its Stable Region Runge-Kutta Method     ode23, ode45 Stiff Problem and A-stable, L-stable BDF(Backward Differentiate Formula) FFT(Fast Fourier Transform) Clenshaw-Curtis Quadrature

HW1 Problem Sol basic Matlab operation machine epsilon   HW2 Problem Sol basic Matlab operation truncation error and computation error   HW3 Problem Sol Hilbert Matrix and Conditioning Number Gauss Elimination Implement mydet.m to calculate determinant Implement myinv.m to calculate inverse matrix Implement mychol.m by Cholesky Algorithm to decompose positive-definite matrix   HW4 Problem Sol Implement Newton's Method, Secant Method and Bisection Method Application of fzero   Midterm Problem Sol   HW5 Problem Sol Minimizer fzerotx and its application Drawing contour Implement minimizer by Newton's Method and BFGS along with line search   HW6 Problem Sol Equidist and Chebyshev Nodes Implement Polynomial Difference Method by Barycentric Formula Implement Spline and PCHIP with periodic boundary condition Drawing approx graphic by Difference Method   HW7 Problem Sol Calculate Finite Difference Coefficients Implement intergration by Gauss Quadrature Implement Simpson Quadrature and analyze its intergration on erf function Solving Trefethen 1st Problem by Aitken's Method   HW8 Problem Sol Calculate stability, consistency and convergency of LMM Analysis on Mildly Stiff Problem by ode23 Implement Runge-Kutta 45 Method Solving ODE system by ode113 Calculate period of pendulum and draw its phase plane   Final Problem Sol