COMPUTING FOR NUMERICAL METHODS USING VISUAL C++
CONTENTS 1 MODELING AND SIMULATION NUMERICAL APPROXIMATION C++ FOR NUMERICAL MODELING MATHEMATICAL MODELING SIMULATION AND ITS VISUALIZATION NUMERICAL METHODS NUMERICAL APPLICATIONS 2 FUNDAMENTAL TOOLS FOR MATHEMATICAL COMPUTING C++ FOR HIGH-PERFORMANCE COMPUTING DYNAMIC MEMORY ALLOCATION MATRIX REDUCTION PROBLEMS MATRIX ALGEBRA ALGEBRA OF COMPLEX NUMBERS NUMBER SORTING SUMMARY PROGRAMMING CHALLENGES 3 NUMERICAL INTERFACE DESIGNS MICROSOFT FOUNDATION CLASSES GRAPHICS DEVICE INTERFACE WRITING A BASIC WINDOWS PROGRAM DISPLAYING TEXT AND GRAPHICS EVENTS AND METHODS STANDARD CONTROL RESOURCES MENU AND FILE I/O KEYBOARD CONTROL MFC COMPATIBILITY WITH NET 3 10 SUMMARY 4 CURVE VISUALIZATION TOOLS FOR VISUALIZATION MYPARSER DRAWING CURVES GENERATING CURVES USING MYPARSER 5 SYSTEMS OF LINEAR EQUATIONS INTRODUCTION EXISTENCE OF SOLUTIONS GAUSSIAN ELIMINATION TECHNIQUES LU FACTORIZATION METHODS ITERATIVE TECHNIQUES NUMERICAL EXERCISES 6 NONLINEAR EQUATIONS BISECTION METHOD FALSE POSITION METHOD NEWTON–RAPHSON METHOD SECANT METHOD FIXED-POINT ITERATION METHOD 7 INTERPOLATION AND APPROXIMATION CURVE FITTING LAGRANGE INTERPOLATION NEWTON INTERPOLATIONS CUBIC SPLINE LEAST-SQUARES APPROXIMATION 8 DIFFERENTIATION AND INTEGRATION NUMERICAL DIFFERENTIATION NUMERICAL INTEGRATION 9 EIGENVALUES AND EIGENVECTORS EIGENVALUES AND THEIR SIGNI?CANCE EXACT SOLUTION AND ITS EXISTENCE POWER METHOD SHIFTED POWER METHOD QR METHOD 10 ORDINARY DIFFERENTIAL EQUATIONS INITIAL-VALUE PROBLEM FOR FIRST-ORDER ODE TAYLOR SERIES METHOD RUNGE–KUTTA OF ORDER 2 METHOD RUNGE–KUTTA OF ORDER 4 METHOD PREDICTOR-CORRECTOR MULTISTEP METHOD SYSTEM OF FIRST-ORDER ODES SECOND-ORDER ODE INITIAL-VALUE PROBLEM FOR SECOND-ORDER ODE 10 10 FINITE-DIFFERENCE METHOD FOR SECOND-ORDER ODE 10 11 DIFFERENTIATED BOUNDARY CONDITIONS 10 13 SUMMARY 11 PARTIAL DIFFERENTIAL EQUATIONS POISSON EQUATION LAPLACE EQUATION HEAT EQUATION WAVE EQUATION PROGRAMMING EXERCISES CHAPTER NUMERICAL APPROXIMATIONS NUMERICAL METHODS IS AN AREA OF STUDY IN MATHEMATICS THAT DISCUSSES THE SOLUTIONS TO VARIOUS MATHEMATICAL PROBLEMS INVOLVING DIFFERENTIAL EQUATIONS, CURVE ?TTINGS, INTEGRALS, EIGENVALUES, AND ROOT ?NDINGS THROUGH APPROXIMATIONS RATHER THAN EXACT SOLUTIONS THIS DISCUSSION IS NECESSARY BECAUSE THE EXACT SOLUTIONS TO THESE PROBLEMS ARE DIF?CULT TO OBTAIN THROUGH THE ANALYTICAL APPROACH FOR EXAMPLE, IT MAY BE WISE TO EVALUATE X SIN X DX, AS THE EXACT SOLUTION CAN BE OBTAINED THROUGH A WELL-KNOWN TECHNIQUE IN CALCULUS CALLED INTEGRATION BY PARTS HOWEVER, IT IS NOT POSSIBLE TO APPLY THE SAME METHOD OR ANY OTHER ANALYTICAL METHOD TO SOLVE E?X 3 SIN X + X2 DX THE GIVEN EQUATION IN THE ABOVE INTEGRAL IS DIF?CULT TO SOLVE AS IT IS NOT SUBJECT TO THE EXACT METHODS DISCUSSED IN ORDINARY CALCULUS THEREFORE, A NUMERICAL METHOD IS NEEDED TO PRODUCE A REASONABLY GOOD APPROXIMATED SOLUTION A GOOD APPROXI- MATED SOLUTION, WHOSE VALUE MAY DIFFER FROM THE EXACT SOLUTION BY SOME FRACTIONS, IS DE?NITELY BETTER THAN NOTHING