Numerical Differentiation (MAT 311) PDF

Numerical Differentiation MAT 311 First order schemes Introduction to Numerical Differentiation ▶ Numerical differentiation approximates the derivative of a function using discrete data points. ▶ Essential when analytical differentiation is impossible or impract...

Numerical Differentiation MAT 311 First order schemes Introduction to Numerical Differentiation ▶ Numerical differentiation approximates the derivative of a function using discrete data points. ▶ Essential when analytical differentiation is impossible or impractical. ▶ Widely used in engineering, physics, and computational sciences. Importance of Numerical Differentiation ▶ Solving differential equations numerically. ▶ Analyzing experimental data to find rates of change. ▶ Implementing numerical methods in simulations. Taylor Series: Foundation of Numerical Differentiation Key Concept Numerical differentiation schemes approximate continuous derivatives using difference quotients. These formulas are primarily derived from Taylor series expansions. Taylor Series Expansion For a function f (x) that is infinitely differentiable at point x: f ′′ (x) 2 f ′′′ (x) 3 f (x + h) = f (x) + f ′ (x)h + h + h + O(h4 ) 2! 3! ▶ Provides systematic approximation of functions near a point ▶ Forms basis for deriving finite difference formulas ▶ Error term O(h4 ) indicates truncation accuracy Forward Difference: Taylor Series Derivation Step 1: Taylor expansion about x f ′′ (x) 2 f ′′′ (x) 3 f (x + h) = f (x) + f ′ (x)h + h + h + O(h4 ) 2! 3! Step 2: Rearrange to make f ′ (x) the subject  ′′ f (x + h) − f (x) f (x) f ′′′ (x) 2  ′ f (x) = − h+ h +O(h3 ) h 2 6 | {z } truncation error f (x + h) − f (x) Forward Difference Approximation: f ′ (x) ≈ h f ′′ (x) with leading truncation error term: E = − h 2 The approximation is first-order accurate, denoted by O(h) since the leading error term is proportional to h Backward Difference: Taylor Series Derivation Step 1: Taylor expansion about x f ′′ (x) 2 f ′′′ (x) 3 f (x − h) = f (x) − f ′ (x)h + h − h + O(h4 ) 2! 3! Step 2: Rearrange to make f ′ (x) the subject  ′′ f (x) − f (x − h) f (x) f ′′′ (x) 2  ′ f (x) = + h− h +... h 2 6 | {z } truncation error f (x) − f (x − h) Backward Difference Approximation: f ′ (x) ≈ h f ′′ (x) with leading truncation error term: E = h 2 The approximation is first-order accurate, denoted by O(h) since the leading error term is proportional to h Central Difference: Taylor Series Derivation Step 1: Taylor expansions about x f ′′ (x) 2 f ′′′ (x) 3 f (x + h) = f (x) + f ′ (x)h + h + h + O(h4 ) 2! 3! f ′′ (x) 2 f ′′′ (x) 3 f (x − h) = f (x) − f ′ (x)h + h − h + O(h4 ) 2! 3! Step 2: Subtract equations to make f ′ (x) while eliminating f ′′ (x)  ′′′ f (x + h) − f (x − h) f (x) 2  ′ f (x) = − h +... 2h 6 | {z } truncation error f (x + h) − f (x − h) Central Difference Approximation: f ′ (x) ≈ 2h f ′′′ (x) 2 with leading truncation error term: E = − h 6 The approximation is second-order accurate, denoted by O(h2 ) since the leading error term is proportional to h2 Three-Point Forward Difference Step 1: Taylor expansions about x f ′′ (x) 2 f ′′′ (x) 3 f (4) (x) 4 f (x + h) = f (x) + f ′ (x)h + h + h + h + O(h5 ) 2! 3! 4! (1) 8f ′′′ (x) 3 16f (4) (x) 4 f (x + 2h) = f (x) + 2f ′ (x)h + 2f ′′ (x)h2 + h + h + O(h5 ) 3! 4! (2) Step 2: Multiply equation (1) by 4 and subtract equation (2) 2f ′′′ (x) 3 2f (4) (x) 4 4f (x + h) = 4f (x) + 4f ′ (x)h + 2f ′′ (x)h2 + h + h + O(h5 ) 3 3 ′′′ (4) 8f (x) 3 16f (x) 4 −f (x + 2h) = −f (x) − 2f ′ (x)h − 2f ′′ (x)h2 − h − h − O(h5 ) 6 24 ′′′ 2f (x) 3 2f (x) 4 (4) 4f (x + h) − f (x + 2h) = 3f (x) + 2f ′ (x)h + h − h + O(h5 ) 3 3 Step 3: Rearrange to make f ′ (x) the subject  ′′′ −f (x + 2h) + 4f (x + h) − 3f (x) f (x)  f ′ (x) = − h2 +O(h3 ) 2h 3 | {z } truncation error Three-Point Forward Difference The Three-Point Forward Difference is give by −f (x + 2h) + 4f (x + h) − 3f (x) f ′ (x) ≈ 2h f ′′′ (x) 2 with leading truncation error term: E = − h The 3 approximation is second-order accurate, denoted by O(h2 ) since the leading error term is proportional to h2 Exercises (Part 1) 1. Second-Order Backward Difference: 3f (x) − 4f (x − h) + f (x − 2h) f ′ (x) ≈ 2h f ′′′ (x) 2 Truncation Error: E = 3 h (Second-order accurate) 2. Third-Order Forward Difference: −11f (x) + 18f (x + h) − 9f (x + 2h) + 2f (x + 3h) f ′ (x) ≈ 6h f (4) (x) 3 Truncation Error: E = 4 h (Third-order accurate) 3. Third-Order Backward Difference: 2f (x − 3h) − 9f (x − 2h) + 18f (x − h) − 11f (x) f ′ (x) ≈ 6h (4) (x) Truncation Error: E = − f 4 h3 (Third-order accurate) Exercises (Part 2) 4. Fourth-Order Central Difference: −f (x + 2h) + 8f (x + h) − 8f (x − h) + f (x − 2h) f ′ (x) ≈ 12h (5) (x) Truncation Error: E = − f 30 h4 (Fourth-order accurate) 5. Four-Point Forward Difference: −25f (x) + 48f (x + h) − 36f (x + 2h) + 16f (x + 3h) − 3f (x + 4h) f ′ (x) ≈ 12h f (5) (x) 4 Truncation Error: E = 5 h (Fourth-order accurate)