THE MINISTRY OF EDUCATION AND SCIENCE OF UKRAINE

DONETSK NATIONAL TECHNICAL UNIVERSITY

WORKBOOK

LABORATORY WORKS ON

“MATHEMATICAL METHODS AND MODELS ”

DONETSK –DonNTU – 2002

THE MINISTRY OF EDUCATION AND SCIENCE OF UKRAINE

DONETSK NATIONAL TECHNICAL UNIVERSITY

WORKBOOK

LABORATORY WORKS ON

“MATHEMATICAL METHODS AND MODELS ”

Approved at the meeting of the

Department “Industrial Enterprise

And City Power Supply”

Minute №3/02 of 27.02.2002.

Approved at the meeting of

The Academic Editorial Board of

DonNTU

Minute № 3 of 03.04.2002 .

DONETSK –DonNTU- 2002

УДК 681.332(07)

Workbook for laboratory works on “Mathematical Methods and Models” (for the students of the speciality 7.090603 Energy Consumption Electrical Systems”) /Staff: S.G. Dzhura, S.V. Shlepnyov, V.V. Yakimishina.

The theoretical data on the computational mathematics methods applied to engineering tasks solving in electrical and power engineering is presented. The task and methodical recommendations to do 16 laboratory works to train students on the methods of computer solving of linear and non-linear equations and their systems, differential equations, methods of numerical integration, function approximation, extreme value search are given.

Compliers: S.G. Dzhura, Associate Professor.

S.V. Shlepnyov, Associate Professor V.V. Yakimishina, Assistant Professor

Reviewer: V. M. Pavlysh, Professor

Inspector: V.I. Chursinov, Associate Professor

Introduction

One of the main current trends of the science and technical advance is the development of methods and means of the information science and computing.

The application of the mathematical methods of engineering task computer solving raises the efficiency of design, parameter computation, research, analysis and synthesis of different technical systems, including those of power supply. As to the mathematics, many energy and electrical engineering tasks add up to solving of algebraic, transcendental and differential equations and their systems, matrix, vector and set operations, table function approximation, functional minimization etc. These tasks can’t always be solved analytically and require numerical method application.

This workbook has tasks and methodical recommendations to the laboratory works necessary to gain the skills of algorithmization, programming, and computer tasks solving with the help of the computational mathematics methods.

The tasks provided in the workbook can be replaced by the analogous ones from the parallel disciplines or coordinated with the SRW topics. This as well as the programming language is to be approved by a teacher.

Laboratory Work 1 computing of polynomial value according to the horner’s method

Purpose of the work: to learn to compute the polynomial values in the most economical way, to gain the programming skills with the application of the user’s functions and sub-programmes.

1.1 Theoretical Data

There is the necessity to compute the functions which look like a polynomial when automated control systems are analyzed and synthesized as well as in the electric circuit theory:

P_n(x)=а₀ xⁿ+а₁ x^n-1+…+а_n-1x+а_n= , (1.1)

Where n – polynomial degree;

=(а₀, а₁, …, а_n)- coefficient vector,

х – independent variable .

The polynomial (1.1) can be converted into:

P_n(x)=(…((( ₀x+ ₁)x+ ₂)x+ ₃)x+…+ _n) . (1.2)

The computing algorithm P_n(x), formed on the basis of the phrase (1.2) is called the Horner’s method.

According to this method the polynomial of the i-order is stated through the polynomial of the (і-1)-order according to the formula

P_i=P_i-1x+ _i. (1.3)

Taking P₀=а₀ and doing the operation (1.3) n times under і=1,2,...,n, the necessary value is obtained.

The Horner’s method is proved to be the most economical algorithm for the general form polynomials as to the number of operations (n additions and n multiplications).

1.2 Tasks

Compute the value of the variable z under x which changes from -1 to + 1 with the step 0.1. The phrases for z computing are given in the table 1.1. In these phrases the functions f₁(x), f₂(x) і f₃(x) are polynomials which differ from each other by the coefficient order and value

For odd variants:

f₁(x)=1.07x⁵-12x⁴-2.8x³+6.3x²+3.7x+4,

f₂(x)=10.1x⁷+37x⁵-15x⁴+8.2x+5.4,

f₃(x)=-23x³+13.6x²+0.5x-1.2.