A Heuristic Method for Solving Polynomial Equations

1984 ◽  
Vol 77 (9) ◽  
pp. 710-714
Author(s):  
R D. Small
Keyword(s):  
Approximate Solution ◽  
Heuristic Method ◽  
Solution Process ◽  
Polynomial Equations ◽  
Numerical Techniques ◽  
Quadratic Formula ◽  
Quadratic Equations ◽  
Insight Into

“When you can see an approximate solution the battle is half over.” This statement is the theme of the following method for solving polynomial equations. When the quadratic formula generates solutions of quadratic equations, we gain little insight into the solution process. A little more insight can be gained by using numerical techniques such as those reviewed by Snover and Spikell (1979). since the convergence to the solution can be observed.

4. Quadratic equations

2015 ◽  
pp. 40-52
Author(s):  
Peter M. Higgins
Keyword(s):  
General Equation ◽  
Solution Process ◽  
Quadratic Equation ◽  
Quadratic Formula ◽  
Quadratic Equations ◽  
Quadratic Approximations

A quadratic equation is one involving a squared term and takes on the form ax2 + bx + c = 0. Quadratic expressions are central to mathematics, and quadratic approximations are extremely useful in describing processes that are changing in direction from moment to moment. ‘Quadratic equations’ outlines the three-stage solution process. Firstly, the quadratic expression is factorized into two linear factors, allowing two solutions to be written down. Next is completing the square, which allows solution of any particular quadratic. Finally, completing the square is applied to the general equation to derive the quadratic formula that allows the three coefficients to be put into the associated expression, which then provides the solutions.

Complete Solution to the Eight-Point Path Generation of Slider-Crank Four-Bar Linkages

2010 ◽  
Vol 132 (8) ◽  
Author(s):  
Hafez Tari ◽  
Hai-Jun Su
Keyword(s):  
Complete Solution ◽  
Solution Process ◽  
Polynomial Equations ◽  
Homotopy Methods ◽  
Degree Polynomial ◽  
Fourth Degree ◽  
Augmented System ◽  
Polynomial Curve ◽  
Four Bar Linkage ◽  
Classical Homotopy

We study the synthesis of a slider-crank four-bar linkage whose coupler point traces a set of predefined task points. We report that there are at most 558 slider-crank four-bars in cognate pairs passing through any eight specified task points. The problem is formulated for up to eight precision points in polynomial equations. Classical elimination methods are used to reduce the formulation to a system of seven sixth-degree polynomials. A constrained homotopy technique is employed to eliminate degenerate solutions, mapping them to solutions at infinity of the augmented system, which avoids tedious post-processing. To obtain solutions to the augmented system, we propose a process based on the classical homotopy and secant homotopy methods. Two numerical examples are provided to verify the formulation and solution process. In the second example, we obtain six slider-crank linkages without a branch or an order defect, a result partially attributed to choosing design points on a fourth-degree polynomial curve.

Computer-Oriented Mathematics: Quadratic Equations—Computer Style

1969 ◽  
Vol 62 (4) ◽  
pp. 305-309
Author(s):  
Walter Koetke ◽  
Thomas E. Kieren
Keyword(s):  
Quadratic Formula ◽  
Quadratic Equations ◽  
Modern Mathematics

ONE of the “old” topics that has been approached in a “new” way in modern mathematics is quadratic equations. No longer do students simply memorize the quadratic formula and do hundreds of exercises using it.

Sharing Teaching Ideas: A Squeeze Play on Quadratic Equations

1982 ◽  
Vol 75 (2) ◽  
pp. 132-136
Keyword(s):  
High School ◽  
Mathematics Teacher ◽  
Quadratic Equation ◽  
Senior Year ◽  
Quadratic Formula ◽  
Mathematics Courses ◽  
Quadratic Equations ◽  
Teaching Ideas

As a mathematics teacher whose present assignment is to teach science, I was somewhat dismayed when my physics class wa unable to solve a nontrivial quadratic equation. These students are all enrolled in senior-year mathematics and had taken all lower level mathematics courses available in our small Western Kansas high school. They charged this inability to having forgotten the quadratic formula. To the e students the quadratic formula is a magic passkey to solving “unfactorable” quadratic equations. On further di scussion, l discovered that they vaguely remembered having heard of the method of completing the square, but they saw no connection between the quadratic formula and that method of solving a quadratic equation. They could solve simple quadratics by hit-and-miss factoring, but that was their only tool with which to attack this problem.

A New Look at Some Old Formulas

1985 ◽  
Vol 78 (1) ◽  
pp. 56-58
Author(s):  
Edward C. Wallace ◽  
Joseph Wiener
Keyword(s):  
Quadratic Formula ◽  
Quadratic Equations ◽  
Alternative Approaches ◽  
New Look

Interest in solving quadratic equations has occupied mathematicians for nearly four thousand years. Indeed, by 2000 b.c. the Babylonians had developed a form of the quadratic formula and a method equivalent to completing the square (Eves 1969; Smith 1951). A number of approaches to the solution of quadratic equations are possible. Let's examine some of these alternative approaches to see what new insights we might discover.

Microcomputer-Assisted Mathematics: Polynomial Equations Revisited

1986 ◽  
Vol 79 (9) ◽  
pp. 732-737
Author(s):  
Jillian C. F. Sullivan
Keyword(s):  
High School ◽  
High School Students ◽  
Numerical Methods ◽  
Polynomial Equations ◽  
School Students ◽  
The Real ◽  
Real Solutions ◽  
Quadratic Equations ◽  
Computational Difficulty

Although solving polynomial equations is important in mathematics, most high school students can solve only linear and quadratic equations. This is because the methods for solving cubic and quartic equations are difficult, and no general methods of solution are available for equations of degree higher than four. However, numerical methods can be used to approximate the real solutions of polynomial equations of any degree. Because they involve a great deal of computation they have not traditionally been taught in the schools. Now that most students have access to calculators and computers, this computational difficulty is easily overcome.

Use of Resultant in the Dimensional Synthesis of Linkage Components: Motion Generation With Prescribed Timing

1992 ◽  
Author(s):  
Chuen-Sen Lin ◽  
Bao-Ping Jia
Keyword(s):  
Angular Displacement ◽  
Solution Process ◽  
Original System ◽  
Polynomial Equations ◽  
Dimensional Synthesis ◽  
Motion Generation ◽  
Degree Polynomial ◽  
Fourth Degree ◽  
Closed Form Solutions ◽  
Resultant Theory

Abstract Resultant theory is applied to derive closed-form solutions for the dimensional synthesis of linkage components for a finite number of precision positions for motion generation with prescribed timing. The solutions are in forms of polynomial equations of the exponential of a single unknown angular displacement. The degree of the derived polynomial depends on the number of links in the linkage component and the number of precision positions to be synthesized for, or the number of compatibility equations. The resultant theory is discussed in detail, and the procedure for the derivation of resultant polynomials is demonstrated. This paper shows that, for the case of two compatibility equations, the solution is a six-degree polynomial. For the case of three compatibility equations, the solution is a fifty-fourth degree polynomial. The Bernshtein formula is applied to check the exact number of solutions of the original system of polynomial equations and to verify the validity of the derived resultant polynomials. An algorithm is also proposed for screening out extra solutions which may be generated through the solution process.

Approaching quadratic equations from a right angle

2017 ◽  
Vol 101 (552) ◽  
pp. 424-438
Author(s):  
King-Shun Leung
Keyword(s):  
Secondary School ◽  
Mathematics Curriculum ◽  
Quadratic Equation ◽  
Bisection Method ◽  
Quadratic Formula ◽  
Secondary School Mathematics ◽  
Quadratic Equations ◽  
Newton Raphson ◽  
Raphson Method ◽  
Comprehensive Discussion

The theory of quadratic equations (with real coefficients) is an important topic in the secondary school mathematics curriculum. Usually students are taught to solve a quadratic equation ax2 + bx + c = 0 (a ≠ 0) algebraically (by factorisation, completing the square, quadratic formula), graphically (by plotting the graph of the quadratic polynomial y = ax2 + bx + c to find the x-intercepts, if any), and numerically (by the bisection method or Newton-Raphson method). Less well-known is that we can indeed solve a quadratic equation geometrically (by geometric construction tools such as a ruler and compasses, R&C for short). In this article we describe this approach. A more comprehensive discussion on geometric approaches to quadratic equations can be found in [1]. We have also gained much insight from [2] to develop our methods. The tool we use is a set square rather than the more common R&C. But the methods to be presented here can also be carried out with R&C. We choose a set square because it is more convenient (one tool is used instead of two).

scholarly journals Construction of Benchmark Problems for Solution of Ordinary Differential Equations

1994 ◽  
Vol 1 (5) ◽  
pp. 403-414 ◽  
Author(s):  
Sangchul Lee ◽  
John L. Junkins
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Numerical Solution ◽  
Solution Process ◽  
Optimal Choice ◽  
Benchmark Problems ◽  
Tuning Parameters ◽  
Forcing Function ◽  
The Relationship

An inverse method is introduced to construct benchmark problems for the numerical solution of initial value problems. Benchmark problems constructed in this fashion have a known exact solution, even though analytical solutions are generally not obtainable. The process leading to the exact solution makes use of an initially available approximate numerical solution. A smooth interpolation of the approximate solution is forced to exactly satisfy the differential equation by analytically deriving a small forcing function to absorb all of the errors in the interpolated approximate solution. Using this special case exact solution, it is possible to directly investigate the relationship between global errors of a candidate numerical solution process and the associated tuning parameters for a given code and a given problem. Under the assumption that the original differential equation is well-posed with respect to the small perturbations, we thereby obtain valuable information about the optimal choice of the tuning parameters and the achievable accuracy of the numerical solution. Five illustrative examples are presented.

Characteristic Deflection Domain for Various Compliant Segment Types, and its Importance in Compliant Mechanism Synthesis and Analysis

2014 ◽  
Author(s):  
Ashok Midha ◽  
Sushrut G. Bapat
Keyword(s):  
Boundary Conditions ◽  
Mechanism Design ◽  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Solution Process ◽  
Displacement Boundary ◽  
Fundamental Understanding ◽  
Rigid Body Model ◽  
Compliant Mechanism Design ◽  
Insight Into

Compliant mechanism design inherently requires certain specified displacement boundary conditions to be satisfied. Obtaining realistic solutions for such problem types often becomes a challenge as the number of displacement boundary condition specifications increases. Typically, related failures are attributed to the numerical nature of the solution process. Little attention has been given to the fundamental understanding of the deformation behavior of flexible continuum with respect to its limits of mobility or reach. This paper strives to provide an insight into this aspect of compliant mechanism design. To assist a designer with the specification of realistic and achievable requirements, the concept of characteristic deflection domain has been proposed in the past. This paper systematically develops the characteristic deflection domain for a variety of compliant segment types. The pseudo-rigid-body model (PRBM) representation is utilized for determining the lower and upper boundaries of the deflection domain. The paper further investigates the mobility characteristics of compliant mechanisms comprised of multiple segment types. Case studies are presented that help exemplify the use of the characteristic deflection domain plots. The results suggest that the number, type, and orientation of the compliant segments have a significant effect on the mobility of compliant mechanisms. Thus, care must be exercised by the designer when specifying free-choices/boundary conditions in compliant mechanisms synthesis and analysis.

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