## Applied Mathematic III

Provides fundamental numerical algorithms including linear computations, Fourier transform, ordinary differential equations, finite element method, random numbers, digital filtering, and digital image processing. Students will write programs on these topics to understand and to use practically the alogorithms. (Dept. Mechanical Eng. / Robotics / Micro System Tech., B3)

 1st Introduction: Analytical solution, Numerical solution 2nd ODE: Canonical forms of ordinary differential equations (ODEs) 3rd ODE: Euler method, Heun method, and Runge-Kutta method 4th ODE: Runge-Kutta-Fehlberg method 5th ODE: Constraint stabilization method (CSM) (1st quiz) 6th Linear equations: Gaussian elimination, LU decomposition 7th Linear equations: Pivot selection, Redundant equations 8th Projection: Minimum error solution, Projection matrix 9th Projection: Gram-Schmidt orthogonalization, QR decomposition 10th Approximation: Piecewise linear approx., Spline approx. 11th FEM: Variational principle in statics, system with constraints (2nd quiz) 12th FEM: 1D finite element method (FEM), 2D finite element method (FEM) 13th Fourier Transform: Discrete fourier transform(DFT) 14th Fourier Transform: Fast fourier transform (FFT) 15th Fourier Transform: Matched filter, Phase-only correlation method

 Evaluation: Final Exam. 60% and Quiz 40%

 Textbook: Numerical Methods for Mechanical Systems ISBN 978-4-339-06094-2 References: Linear Algebra and Its Applications Gilbert Strang    Thomson Learning     ISBN 0-15-551005-3

Sample programs

• Basic computations (linear computation, numerical integral, solution of equation)  Readme Readme Makefile Makefile Header file opt.h Linear computation mat.c mat-t0.c Numerical integral int.c int-t0.c int-t1.c Solution of equation eqs.c eqs-t0.c

• Ordinary differential equation (Runge-Kutta method)  Readme Readme Header file RK.h Runge-Kutta method rungekutta.c Makefile Makefile

 Free vibration freevibration.c  freevibration.plt Forced vibration twoorder.c      twoorder.plt ODF of the first order oneorder.c      oneorder.plt sample rungekutta-t0.c  rungekutta-t0.plt sample (simple vibration) rungekutta-t1.c  rungekutta-t1.plt sample (vibration of multiple mass particles) rungekutta-t2.c  rungekutta-t2.plt

• Optimization (quasi-Newton method, Nelder-Mead method, multiplier method)  Readme Readme Makefile Makefile Header file opt.h Quasi-Newton method bfgs.c bfgs-t0.c bfgs-t1.c Nelder-Mead method neldermead.c neldermead-t0.c neldermead-t1.c Multiplier method with quasi-Newton multi.c multi-t0.c multi-t1.c Multiplier method with Nelder-Mead multisimp.c multisimp-t0.c

• Digital image processing  Makefile Makefile Header file vision.h File I/O pgmio.c imagewrite-t.c Digital image filter imagefilter.c imagefilter-t.c photo0.pgm Geometric transformation geometry.c imagemove-t.c object.pgm

• Random numbers and digital filtering  Random Numbers (HP) Mersenne Twister (HP) Readme Readme Header file filter.h Random numbers random.c Digital fintering filter.c Gnuplot pva.plt Makefile Makefile

• Linear Programming (Simplex method, Two-phase method)  Readme Readme Makefile Makefile Header file opt.h Simplex method lp.c lp-t0.c   lp-t1.c   lp-t-0.dat   lp-t-1.dat Two-phase method feas.c feas-t0.c   feas-t1.c   feas-t1-0.dat   feas-t1-2.dat feas-t2.c   feas-t3.c   feas-t3-0.dat   feas-t3-1.dat   feas-t3-2.dat

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