Numerical Methods. Equations and Partial Fractions. Jaesung Lee
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1 Numerical Methods Equations and Partial Fractions Jaesung Lee
2 Solving linear equations
3 Solving linear equations Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or more unknown quantities which you will be required to find. We consider a particular type of equation which contains a single unknown quantity, and is known as a linear equation. 3
4 Solving linear equations Linear equations A linear equation is an equation of the form + = 0 where 0 where and are known numbers and represents an unknown quantity which we must find. 4
5 Solving linear equations Linear equations In the equation ax + b = 0, the number a is called the coefficient of x, and the number b is called the constant term. The following are examples of linear equations = 0, = 0, 3 = 0 Note that the unknown,, appears only to the first power, that is as, and not as,, / etc. Linear equations often appear in a nonstandard form, and also different letters are sometimes used for the unknown quantity. For example, 2 = + 1, 3 7 = 17, 1 = 3 are all examples of linear equations. 5
6 Solving quadratic equations
7 Solving quadratic equations Introduction A quadratic equation is one which can be written in the form + + = 0 where, and are numbers and is the unknown whose value(s) we wish to find. Exact definition of the quadratic equations is + + = 0 where 0 7
8 Solving quadratic equations Solution by factorization It may be possible to solve a quadratic equation by factorization, although you should be aware that not all quadratic equations can be easily factorized. For example, + 5 = 0 can be factorized as + 5 = 0 so that = 0 and = 5 are the two solutions. 8
9 Solving quadratic equations Completing the square For example, if we want to solve the equation = 0 by completing the square, then 1) First of all just consider + 6 and rewrite as = = 0 2) Simply, + 3 = 7 3) Taking the square root of both sides gives + 3 = ± 7 = 3 ± 7 4) The two solutions are = and =
10 Solving quadratic equations Solution by formula When it is difficult to factorize a quadratic equation, it may be possible to solve it using a formula which is used to calculate the roots. The formula is obtained by completing the square in the general quadratic + +. We proceed by removing the coefficient of : + + = + + =
11 Solving quadratic equations Solution by formula Thus the solution of + + = 0 is the same as the solution to + + = 0 So, solving this equation leads to = ± + 11
12 Solving quadratic equations Solution by formula Simplifying = ± + further, we obtain = and = 12
13 Solving quadratic equations Geometrical description of quadratics We can plot a graph of the function = + + (given values of, and ). 13
14 Solving quadratic equations Geometrical description of quadratics If the graph crosses the horizontal axis it will do so when = 0, and so the coordinates at such points are solutions of + + = 0. 14
15 Solving quadratic equations Geometrical description of quadratics Depending on the sign of a and of the nature of the solutions there are essentially just six different types of graph that can occur. These are displayed in below figure. 15
16 Solving polynomial equations
17 Solving polynomial equations Introduction Linear and quadratic equations are members of a class of equations called polynomial equations. These have the general form: = 0 in which is a variable and,,...,,, are given constants. Also must be a positive integer. Examples include = 0 and + = 0. 17
18 Solving polynomial equations Multiplying polynomials together Let us consider what happens when two polynomials are multiplied together. For example, ( + 1)(3 2) is the product of two first degree polynomials. Expanding the brackets, we obtain ( + 1)(3 2) = which is a second degree polynomial. 18
19 Solving polynomial equations Multiplying polynomials together In general, we can regard a second degree polynomial, or quadratic, as the product of two first degree polynomials, provided that the quadratic can be factorized. On the other hand, = is a third degree, or cubic, polynomial which is thus the product of a linear polynomial and a quadratic polynomial. 19
20 Solving polynomial equations Solving polynomial equations when one solution is known We know a formula which can be used to solve quadratic equations. Unfortunately when dealing with equations of higher degree no simple formulae exist. If one of the roots can be spotted we can sometimes find the others by the method shown in the next example. 20
21 Solving polynomial equations Solving polynomial equations when one solution is known Let the polynomial expression be denoted by. Verify that = 4 is a solution of the equation = 0. Hence find the other solutions. We substitute = 4 into the polynomial expression : = = 0 So, when = 4, the left-hand side equals zero. 21
22 Solving polynomial equations Solving polynomial equations when one solution is known Let the polynomial expression be denoted by. Verify that = 4 is a solution of the equation = 0. Hence find the other solutions. Knowing that = 4 is a root we can state that 4 must be a factor of. Therefore, can be re-written as a product of a linear and a quadratic term: = = 4 ( ) 22
23 Solving polynomial equations Solving polynomial equations when one solution is known Let the polynomial expression be denoted by. Verify that = 4 is a solution of the equation = 0. Hence find the other solutions Substitute 23
24 Solving polynomial equations Solving polynomial equations when one solution is known Let the polynomial expression be denoted by. Verify that = 4 is a solution of the equation = 0. Hence find the other solutions. So the given equation can be written = = = 0 In this form, we see that 4 = 0 or = 0 24
25 Solving polynomial equations Solving polynomial equations when one solution is known Let the polynomial expression be denoted by. Verify that = 4 is a solution of the equation = 0. Hence find the other solutions. Using the formula, we are able to find the roots of = 0 = ± = ± Thus, = 4, = and =. 25
26 Partial Fractions
27 Partial fractions Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. For example it can be shown that has the same value as + for any value of. In this case, we say that is identically equal to + the partial fractions of and that are and. 27
28 Partial fractions Introduction The ability to express a fraction as its partial fractions is particularly useful in the study of Laplace transforms, of z-transforms, in Control Theory and in integration. We will see how partial fractions are found. 28
29 Partial fractions Proper and improper fractions We find that an algebraic fraction appears in the form algebraic fraction = where both numerator and denominator are polynomials. For example,,, and. 29
30 Partial fractions Proper and improper fractions The degree of the numerator, say, is the highest power occurring in the numerator. The degree of the denominator, say, is the highest power occurring in the denominator. If > the fraction is said to be proper.,, and 30
31 Partial fractions Proper and improper fractions If the fraction is said to be improper; the first and second expressions are examples of this type. Before calculating the partial fractions of an algebraic fraction, it is important to decide the fraction is proper or improper.,, and 31
32 Partial fractions Proper fractions with linear factors Firstly we describe how to calculate partial fractions for proper fractions where the denominator may be written as a product of linear factors. The steps are as follows: Factorize the denominator. Each factor will produce a partial fraction. A factor such as will produce a partial fraction of the form where is an unknown constant. 32
33 Partial fractions Proper fractions with linear factors In general, a linear factor + will produce a partial fraction. The unknown constants for each partial fraction may be different and so we will call them,, and so on. Evaluate the unknown constants by equating coefficients or using specific values of. The sum of the partial fractions is identical in value to the original algebraic fraction for any value of. 33
34 Partial fractions Proper fractions with linear factors (Example) Express in terms of partial fractions. 34
35 Partial fractions Proper fractions with quadratic factors Sometimes a denominator is factorized producing a quadratic term which cannot be factorized into linear factors. One such quadratic factor is This factor produces a partial fraction of the form. In general, a quadratic factor of the form + + produces a single partial fraction of the form. 35
36 Partial fractions Proper fractions with quadratic factors (Example) Express as partial fractions. Note that the quadratic factor cannot be factorized further. 36
37 Partial fractions Improper fractions When calculating the partial fractions of improper fractions, an extra polynomial is added to any partial fractions that would normally arise. The added polynomial has degree where is the degree of the denominator and is the degree of the numerator. Recall that a polynomial of degree 0 is a constant, say, a polynomial of degree 1 has the form +, a polynomial of degree 2 has the form + +, and so on. 37
38 Partial fractions Improper fractions (Example) Express as partial fractions. 38
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