Numerical Integration Contd Numerical Integration Rectangular Rule fi12

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Numerical Integration (Contd) •

Numerical Integration (Contd) •

Numerical Integration: Rectangular Rule fi+1/2 fi f 1 fi-1 x 1 xi-1 fi+2 fi+1

Numerical Integration: Rectangular Rule fi+1/2 fi f 1 fi-1 x 1 xi-1 fi+2 fi+1 fn f 0 a = x 0 • xi xi+1/2 xi+1 xi+2 b = xn

Numerical Integration: Rectangular Rule fi+1/2 fi f 1 fi-1 x 1 xi-1 fi+2 fi+1

Numerical Integration: Rectangular Rule fi+1/2 fi f 1 fi-1 x 1 xi-1 fi+2 fi+1 fn f 0 a = x 0 • xi xi+1/2 xi+1 xi+2 b = xn

Numerical Integration: Trapezoidal Rule fi f 1 fi-1 x 1 xi-1 fi+2 fi+1 fn

Numerical Integration: Trapezoidal Rule fi f 1 fi-1 x 1 xi-1 fi+2 fi+1 fn f 0 a = x 0 • xi xi+1 xi+2 b = xn

Numerical Integration: Trapezoidal Rule •

Numerical Integration: Trapezoidal Rule •

Numerical Integration: Simpson’s Rules fi f 1 fi-1 x 1 xi-1 fi+2 fi+1 fn

Numerical Integration: Simpson’s Rules fi f 1 fi-1 x 1 xi-1 fi+2 fi+1 fn f 0 a = x 0 • xi xi+1 xi+2 b = xn

Numerical Integration: Simpson’s Rules •

Numerical Integration: Simpson’s Rules •

Numerical Integration: Simpson’s Rules •

Numerical Integration: Simpson’s Rules •

Numerical Integration: Simpson’s Rules •

Numerical Integration: Simpson’s Rules •

Numerical Integration: Simpson’s Rules fi f 1 fi-1 x 1 xi-1 fi+2 fi+1 fn

Numerical Integration: Simpson’s Rules fi f 1 fi-1 x 1 xi-1 fi+2 fi+1 fn f 0 a = x 0 • xi xi+1 xi+2 b = xn

Numerical Integration: Simpson’s Rules •

Numerical Integration: Simpson’s Rules •

Numerical Integration: Simpson’s Rules •

Numerical Integration: Simpson’s Rules •

Numerical Integration: Simpson’s Rules fi f 1 fi-1 x 1 xi-1 fi+2 fi+1 fn

Numerical Integration: Simpson’s Rules fi f 1 fi-1 x 1 xi-1 fi+2 fi+1 fn f 0 a = x 0 • xi xi+1 xi+2 b = xn

Numerical Integration •

Numerical Integration •

Romberg Integration •

Romberg Integration •

Gauss Quadrature •

Gauss Quadrature •

(a) Graphical depiction of Trapezoidal Rule (b) Improved integral estimate by taking the area

(a) Graphical depiction of Trapezoidal Rule (b) Improved integral estimate by taking the area under the straight line passing through two intermediate points. By positioning these points wisely, the positive and negative errors are balanced, and an improved integral estimate results Source: Chapra and Canale, pg 641 (2012)

Gauss Quadrature •

Gauss Quadrature •

Gauss Quadrature •

Gauss Quadrature •

Gauss Quadrature •

Gauss Quadrature •

Gauss Quadrature •

Gauss Quadrature •

Gauss-Legendre Quadrature •

Gauss-Legendre Quadrature •

Gauss-Legendre Quadrature: Example •

Gauss-Legendre Quadrature: Example •

Gauss-Legendre Quadrature: Example •

Gauss-Legendre Quadrature: Example •

Numerical Integration: Example •

Numerical Integration: Example •

Numerical Integration: Example •

Numerical Integration: Example •

Numerical Integration: Example •

Numerical Integration: Example •

Numerical Integration: Example • I

Numerical Integration: Example • I

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