Numerical Analysis Lecture 2

Numerical Analysis Lecture 2 Topics

• Solution of Non-Linear Equations: Methods for finding solutions to equations that are not linear.
• Solution of Linear Systems of Equations: Techniques for solving systems of linear equations.
• Approximation of Eigenvalues: Methods for estimating the eigenvalues of a matrix.
• Interpolation and Polynomial Approximation: Creating polynomial approximations to fit given data points.
• Numerical Differentiation: Finding the derivative of a function using numerical methods.
• Numerical Integration: Determining the definite integral of a function using numerical methods.
• Numerical Solution of Ordinary Differential Equations: Finding approximate solutions to ordinary differential equations.
• Errors in Computations: Identifying and categorizing errors in numerical computations.

Error Types

• Inherent Errors: Errors present in the problem statement itself, before any computation is done. They stem from the simplified assumptions made in the mathematical model. These errors occur due to approximations in data or simplified mathematical representations of real-world problems.
• Local Round-Off Errors: Errors introduced by the finite word length of computers. Computers can only store a limited number of digits for numbers, leading to rounding errors during computations and data storage. Errors happen when the decimal-based numbers are stored in computers' binary systems.
• Local Truncation Errors: Errors introduced when an infinite series is truncated, keeping only a finite number of terms. Usually involved in approximating continuous functions like cos x. This is because the Taylor series of functions are infinite in nature.

Error Measurement

• Absolute Error: The absolute difference between the true value and the computed value.
• Relative Error: The absolute error divided by the true value. Also called the percentage error.

Local Truncation Error (TE)

• Method & Formula: Using Taylor series expansion and truncating the expansion to a finite number of terms results in an error. In the context of cos x , there is a truncation error formula. This formula involves a term of the form x²ⁿ⁺²/(2n+2)!, where x is the input, and n defines how many terms are being retained. The formula also depicts a decreasing trend in the truncation error as n increases.
• Independence: The Local Truncation Error is independent of the specific computer being used.
• Finding Appropriate Term Count: Determining the necessary number of terms to reach desired accuracy in approximating cos x for example. This can be done using the formula for TE and the required accuracy of the final result.