5. Introduction to Numerical
Computation
Numerical computation is the study of
algorithms that use numerical approximation for
mathematical analysis. It is a fundamental aspect
of computer science and is widely used in
various fields such as engineering, physics, and
finance.
6. Importance
Importance ofApproximation in Numerical Computation:
Efficiency:
• Numerical approximation allows for efficient
computation of complex complex mathematical
functions and equations, saving time and resources.
Real-world Applications:
• Approximation methods enable the solution of
practical problems that problems that may not have
exact analytical solutions.
Algorithm Design:
• Many algorithms and computational models rely on
approximation to approximation to achieve accurate
and reliable results
7. Truncation Errors
What are Truncation Errors?
Truncation errors result from an incomplete
representation of an infinite process, often arising in
arising in numerical integration and differentiation.
differentiation.
Example:
An example of truncation error is the use of a Taylor
series to approximate a function, where the series is
truncated after a finite number of terms.
8. Discretization:
When continuous problems are discretized for
numerical solutions, errors can occur due to
the approximation of continuous functions
functions or systems.
Algorithm Complexity:
Complex algorithms with multiple iterations
or recursive steps can introduce significant
truncation errors. errors.
Sources of Truncation Errors
9. Impact of Truncation Errors
Impact of Truncation Errors onAccuracy and and
Precision:
Accuracy:
Truncation errors can lead to inaccuracies in the the computed values,
affecting the reliability of of numerical results.
Precision:
These errors may also reduce the precision of the calculations,
influencing the level of detail in the obtained solutions.
10. Rounding Errors
What are Rounding Errors?
Rounding errors occur when numbers are approximated to a specific
precision, leading to discrepancies between the true value and its
rounded representation.
Example:
An example of rounding error is the conversion of a conversion of a real
number to a floating-point point representation, resulting in a slight loss
of accuracy.
11. Sources of Rounding Errors
Finite Precision Arithmetic
The use of finite precision arithmetic in computer systems can introduce
rounding rounding errors, especially when dealing with with irrational or
non-terminating numbers. numbers.
Computational Iterations
Iterations in numerical algorithms, particularly particularly those
involving division or multiplication, can lead to cumulative rounding
errors.
14. Conclusion and Key Takeaways
Emphasis:
Numerical computation is fundamental in modern
scientific and engineering disciplines, emphasizing
the need for error-free computation.
Continuous Improvement:
Unceasing efforts in error minimization lead to
advancements in numerical computation, benefiting
benefiting numerous fields of study.
