Numerical Methods: Procedures and Applications, Slides of Mathematical Methods for Numerical Analysis and Optimization

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Introduction to Numerical Methods

Mathematical Procedures

Mathematical Procedures

• Nonlinear Equations
• Differentiation
• Simultaneous Linear Equations
• Curve Fitting
  • Interpolation
  • Regression
• Integration
• Ordinary Differential Equations
• Other Advanced Mathematical Procedures:
  • Partial Differential Equations
  • Optimization
  • Fast Fourier Transforms

Nonlinear Equations

How much of the floating ball is under the water?

f ( x ) = x^3 − 0. 165 x^2 + 3. 993 × 10 −^4 = 0

Differentiation

. t

t

v(t) 98

2200 ln 4

4

^ −

× −

×

What is the acceleration at t=7 seconds?

dt

dv a =

Simultaneous Linear Equations

Time (s) 5 8 12

Vel (m/s) 106 177 600

Find the velocity profile, given

v ( t ) = at^2 + bt + c , Three simultaneous linear equations 25 a + 5 b + c = 106

5 ≤ t ≤ 12

64 a + 8 b + c = 177

144 a + 12 b + c = 600

Interpolation

Time (s) 5 8 12 Vel (m/s) 106 177 600

What is the velocity of the rocket at t=7 seconds?

Regression (cont)

Integration

fluid

room

T

T

D D α dT

Finding the diametric contraction in a steel shaft when dipped in liquid nitrogen.