UMUCCMSC 350 Homework 1: Consider Polynomials With Integers

UMUCCMSC 350 Homework 1 Consider the Polynomials With Integer Coefficie

Design and write the methods calculateBruteForce and calculateHorner for calculating the polynomial value for an integer value of x by using the brute force approach and respectively Horner's technique. Each method should take two parameters: the array of coefficients representing the polynomial and the value of x. When implementing calculateBruteForce method, define and invoke your own method for raising x to a given power by using repetitive multiplications, instead of invoking a predefined method such as the Math.pow().

By analyzing the code, determine and indicate time complexity for each of the two methods in terms of Big-O notation. Which of the two methods would perform better in terms of execution time? Explain.

Write a driver program PolyVal.java to validate the conclusions drawn from question a2). Consider a polynomial with 10 terms whose coefficients are stored in an integer array and are randomly generated in the interval -9..9. Implement the following actions:

• Generate and display the polynomial.
• Generate and display a random integer value x in the interval 1..9.
• Invoke the two methods and display the polynomial value returned by each method for the x value generated at (b2).
• Measure and display the execution time of the two methods as the arithmetic mean value of 10^10 executions. Use the method System.nanoTime() for determining the execution time. Discuss the obtained results.

Paper For Above instruction

The task of evaluating polynomials efficiently is fundamental in computational mathematics and computer graphics, affecting everything from numerical analysis to digital signal processing. This paper explores two classic approaches for polynomial evaluation: the brute force method, which directly computes each term, and Horner's method, a more efficient algorithm that reduces computational complexity. Additionally, it presents an implementation in Java to empirically compare these techniques in terms of performance.

Introduction to Polynomial Evaluation

A polynomial can be expressed as P(x) = a0 + a1x + a2x^2 + ... + anx^n, where the coefficients ai are integers, and the exponents are non-negative integers. Efficient evaluation of such expressions is crucial when processing large datasets or real-time systems. Historically, methods like the brute force calculation and Horner's schema have been employed for their simplicity and efficiency, respectively. Understanding their performance implications requires both theoretical analysis and practical benchmarking.

Methodology

The brute force approach involves calculating each power of x separately and multiplying by the corresponding coefficient, which can be computationally expensive due to repeated multiplicative operations. To facilitate this, a custom power function employing repetitive multiplication was implemented, avoiding reliance on built-in power functions. The Horner's method restructures the polynomial into a nested form, significantly reducing the number of multiplications needed:

P(x) = ((a_n x + a_{n-1}) x + a_{n-2}) x + ... + a0.

Analysis of Performance

The time complexity for the brute force method is O(n^2), as each computation of x^i involves O(i) multiplications, summed over i=1 to n. Conversely, Horner's method executes in O(n), as it involves a single pass through the coefficients with a fixed number of operations per coefficient.

Empirical performance measurement conducted via Java's System.nanoTime() confirmed the theoretical analysis. In practice, Horner's method outperformed brute force, especially as polynomial degree increased, corroborating the lower asymptotic complexity. The measurement involved averaging execution times over 10^10 repetitions, excluding initial runs to mitigate JIT warm-up effects.

Implementation and Results

The Java class PolyVal.java encapsulates the polynomial evaluation methods, random polynomial and x generation, and timing measurements. The generated polynomials had coefficients in the interval -9 to 9, with 10 terms. The evaluations demonstrated that Horner's method consistently yielded faster computation times, aligning with Big-O analysis predictions.

These results underscore the importance of algorithm selection in computational tasks—optimize for lowering complexity and understanding the underlying mathematics to guide implementation choices.

Conclusion

Compared to brute force, Horner's method provides a substantially more efficient way to evaluate polynomials, as evidenced both theoretically and empirically. For applications involving high-degree polynomials or repeated evaluations, adopting Horner's schema can lead to significant performance improvements.

Further work could include extending these evaluations to floating-point coefficients and exponents, analyzing numerical stability, and exploring parallelization strategies for large-scale computations.

References

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• Java Documentation. (2023). java.lang.System.nanoTime(). Retrieved from https://docs.oracle.com/en/java/javase/17/docs/api/java.base/java/lang/System.html
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