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Questions and Answers
What happens to the truncation error if more terms are included in the series expansion for cos x?
What happens to the truncation error if more terms are included in the series expansion for cos x?
When truncating the series for cos x, what does the truncation error depend on?
When truncating the series for cos x, what does the truncation error depend on?
How many terms in the series expansion are required to compute cos x accurately to five significant digits?
How many terms in the series expansion are required to compute cos x accurately to five significant digits?
Which mathematical operation is applied to determine the validity of the truncation error condition?
Which mathematical operation is applied to determine the validity of the truncation error condition?
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What is the upper limit of the truncation error when approximating cos x using the series?
What is the upper limit of the truncation error when approximating cos x using the series?
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What is the primary cause of inherent errors in numerical computations?
What is the primary cause of inherent errors in numerical computations?
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How is local round-off error created in a computer system?
How is local round-off error created in a computer system?
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Which of the following best describes local truncation error?
Which of the following best describes local truncation error?
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What is the formula for calculating relative error?
What is the formula for calculating relative error?
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What is the maximum number of bits in a computer system that directly affects the local round-off error?
What is the maximum number of bits in a computer system that directly affects the local round-off error?
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Which type of error is committed when converting binary results back into decimal form for presentation?
Which type of error is committed when converting binary results back into decimal form for presentation?
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Which of these is NOT a source of error in numerical computations?
Which of these is NOT a source of error in numerical computations?
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If the computed value is 0.76245 and the true value is 0.7625, what is the absolute error?
If the computed value is 0.76245 and the true value is 0.7625, what is the absolute error?
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Study Notes
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)
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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
nincreases. - 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.
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Description
Explore the key topics in Numerical Analysis from Lecture 2, including methods for solving non-linear equations, linear systems, and approximating eigenvalues. Delve into numerical differentiation, integration, and the solution of ordinary differential equations. Additionally, learn about errors in computations and their types.
