Symbolics Analysis and Experiments in the Evaluation of Integrals

The evaluation of integrals is a fundamental problem in mathematics. Integrals arise in a wide variety of applications, such as physics, engineering, and finance. There are two main approaches to evaluating integrals: symbolic integration and numerical integration.

Symbolic integration involves finding an exact expression for the integral in terms of elementary functions. This approach is often preferred when the integral is relatively simple. However, symbolic integration can be difficult or impossible for more complex integrals.

Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals

by George Boros

4.5 out of 5

Language	:	English
File size	:	5096 KB
Screen Reader	:	Supported
Print length	:	322 pages
X-Ray for textbooks	:	Enabled

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Numerical integration involves approximating the integral using a numerical method. This approach is often used when the integral is too complex to evaluate symbolically. Numerical integration methods can be very accurate, but they can also be slow and computationally expensive.

In this paper, we present a hybrid approach to evaluating integrals that combines symbolic and numerical techniques. Our approach uses symbolic integration to simplify the integral as much as possible, and then uses numerical integration to evaluate the remaining integral. This approach can often be much faster and more accurate than either symbolic integration or numerical integration alone.

Symbolic Integration

Symbolic integration is the process of finding an exact expression for the integral in terms of elementary functions. This can be done using a variety of techniques, such as integration by parts, substitution, and trigonometric identities.

Symbolic integration can be very difficult for complex integrals. However, there are a number of computer algebra systems that can perform symbolic integration. These systems can be very helpful for simplifying integrals and finding exact solutions.

Numerical Integration

Numerical integration is the process of approximating the integral using a numerical method. There are a variety of numerical integration methods, such as the trapezoidal rule, the Simpson's rule, and the Gaussian quadrature.

Numerical integration methods can be very accurate, but they can also be slow and computationally expensive. The accuracy of a numerical integration method depends on the number of points that are used to approximate the integral. The more points that are used, the more accurate the approximation will be. However, using more points also increases the computational cost of the integration.

Hybrid Approach

Our hybrid approach to evaluating integrals combines symbolic and numerical techniques. We first use symbolic integration to simplify the integral as much as possible. This often involves using integration by parts, substitution, and trigonometric identities. Once the integral has been simplified, we then use numerical integration to evaluate the remaining integral.

This approach can often be much faster and more accurate than either symbolic integration or numerical integration alone. Symbolic integration can be used to simplify the integral and reduce the number of points that are needed for numerical integration. This can lead to a significant improvement in speed and accuracy.

Experiments

We conducted a series of experiments to compare the performance of our hybrid approach to the performance of symbolic integration and numerical integration alone. We used a variety of integrals, including simple integrals, complex integrals, and integrals with singularities.

Our results showed that our hybrid approach was significantly faster and more accurate than either symbolic integration or numerical integration alone. For simple integrals, our hybrid approach was able to achieve the same accuracy as symbolic integration, but much faster. For complex integrals, our hybrid approach was able to achieve much higher accuracy than numerical integration, but with only a modest increase in computational cost.

Our hybrid approach to evaluating integrals combines the strengths of symbolic and numerical techniques. This approach can often be much faster and more accurate than either symbolic integration or numerical integration alone. We believe that our approach has the potential to be a valuable tool for scientists and engineers in a wide variety of fields.

Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals

by George Boros

4.5 out of 5

Language	:	English
File size	:	5096 KB
Screen Reader	:	Supported
Print length	:	322 pages
X-Ray for textbooks	:	Enabled

FREE