Newton's method for solving nonlinear equations c. Solving systems of nonlinear steady-state equations using the Newton-Raphson method

Newton's method (also known as the tangent method) is an iterative numerical method for finding the root (zero) of a given function. The method was first proposed by the English physicist, mathematician and astronomer Isaac Newton (1643-1727), under whose name it became famous.

The method was described by Isaac Newton in the manuscript De analysi per aequationes numero terminorum infinitas (lat. .About analysis by equations of infinite series), addressed in 1669 to Barrow, and in the work De metodis fluxionum et serierum infinitarum (Latin: The method of fluxions and infinite series) or Geometria analytica ( lat.Analytical geometry) in the collected works of Newton, which was written in 1671. However, the description of the method differed significantly from its current presentation: Newton applied his method exclusively to polynomials. He did not calculate successive approximations of x n, but a sequence of polynomials and as a result obtained an approximate solution of x.

The method was first published in the treatise Algebra by John Wallis in 1685, at whose request it was briefly described by Newton himself. In 1690, Joseph Raphson published a simplified description in his work Analysis aequationum universalis (lat. General analysis equations). Raphson viewed Newton's method as purely algebraic and limited its use to polynomials, but he described the method in terms of successive approximations x n instead of the more difficult to understand sequence of polynomials used by Newton.

Finally, in 1740, Newton's method was described by Thomas Simpson as a first-order iterative method for solving nonlinear equations using the derivative as presented here. In the same publication, Simpson generalized the method to the case of a system of two equations and noted that Newton's method can also be applied to solve optimization problems by finding the zero of the derivative or gradient.

In accordance with this method, the task of finding the root of a function is reduced to the task of finding the point of intersection with the x-axis of the tangent plotted to the graph of the function.

Fig.1 . Function change graph

A tangent line drawn at any point to the graph of a function is determined by the derivative of this function at the point under consideration, which in turn is determined by the tangent of the angle α (). The point of intersection of the tangent with the abscissa axis is determined based on the following relationship in right triangle: tangent of anglein a right triangle is determined by the ratio of the opposite side to the adjacent side of the triangle. Thus, at each step, a tangent to the graph of the function is constructed at the point of the next approximation . Point of intersection of the tangent with the axis Ox will be the next approach point. In accordance with the method under consideration, calculating the approximate value of the root oni-iterations are carried out according to the formula:

The slope of the straight line is adjusted at each step in the best possible way, however, you should pay attention to the fact that the algorithm does not take into account the curvature of the graph and, therefore, during the calculation process it remains unknown in which direction the graph may deviate.

The condition for the end of the iterative process is the fulfillment of the following condition:

Where ˗ permissible error in determining the root.

The method has quadratic convergence. The quadratic rate of convergence means that the number of correct signs in the approximation doubles with each iteration.

Mathematical justification

Let a real function be given, which is defined and continuous in the area under consideration. It is necessary to find the real root of the function in question.

The derivation of the equation is based on the method simple iterations, according to which the equation is reduced to an equivalent equation for any function. Let us introduce the concept of a contraction mapping, which is defined by the relation .

For the best convergence of the method, the condition must be satisfied at the point of the next approximation. This requirement means that the root of the function must correspond to the extremum of the function.

Derivative of the contraction mapis defined as follows:

Let us express the variable from this expressionsubject to the previously accepted statement that when it is necessary to ensure the condition . As a result, we obtain an expression for defining the variable:

Taking this into account, the compression function is as follows:

Thus, the algorithm for finding a numerical solution to the equation is reduced to an iterative calculation procedure:

Algorithm for finding the root of a nonlinear equation using the method

1. Set the starting point of the approximate value of the root of the function, as well as the calculation error (small positive number) and the initial iteration step ().

2. Calculate the approximate value of the root of the function in accordance with the formula:

3. We check the approximate value of the root for the specified accuracy, in the case of:

If the difference between two successive approximations becomes less than the specified accuracy, then the iterative process ends.

If the difference between two successive approximations does not reach the required accuracy, then it is necessary to continue the iterative process and go to step 2 of the algorithm under consideration.

Example of solving equations

by methodNewton for an equation with one variable

As an example, consider solving a nonlinear equation using the methodNewton for an equation with one variable. The root must be found with accuracy as a first approximation.

Option for solving a nonlinear equation in a software packageMathCADpresented in Figure 3.

The calculation results, namely the dynamics of changes in the approximate value of the root, as well as the calculation errors depending on the iteration step, are presented in graphical form (see Fig. 2).

Fig.2. Calculation results using Newton's method for an equation with one variable

To ensure the specified accuracy when searching for an approximate value of the root of the equation in the range, it is necessary to perform 4 iterations. At the last iteration step, the approximate value of the root of the nonlinear equation will be determined by the value: .

Fig.3 . Program listing inMathCad

Modifications of Newton's method for an equation with one variable

There are several modifications of Newton's method that are aimed at simplifying the computational process.

Simplified Newton's method

In accordance with Newton's method, it is necessary to calculate the derivative of the function f(x) at each iteration step, which leads to an increase in computational costs. To reduce the costs associated with calculating the derivative at each calculation step, you can replace the derivative f’(x n) at point x n in the formula with the derivative f’(x 0) at point x 0. In accordance with this calculation method, the approximate value of the root is determined by the following formula:Modified Newton's method

Newton's difference method

As a result, the approximate value of the root of the function f(x) will be determined by the expression of Newton’s difference method:

Newton's two-step method

In accordance with Newton's method, it is necessary to calculate the derivative of the function f(x) at each iteration step, which is not always convenient and sometimes practically impossible. This method allows the derivative of a function to be replaced by a difference ratio (approximate value):

As a result, the approximate value of the root of the function f(x) will be determined by the following expression:

Where

Fig.5 . Newton's two-step method

The secant method is a two-step method, that is, a new approximationdetermined by the two previous iterations And . The method must specify two initial approximations And . The convergence rate of the method will be linear.

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2. Newton's method for solving systems of nonlinear equations.

This method has much faster convergence than the simple iteration method. Newton’s method for the system of equations (1.1) is based on the use of function expansion

, Where
(2.1)

in the Taylor series, with terms containing the second or more high orders derivatives are discarded. This approach allows the solution of one nonlinear system (1.1) to be replaced by the solution of a number of linear systems.

So, we will solve system (1.1) by Newton’s method. In region D, choose any point
and call it the zero approximation to the exact solution of the original system. Now let us expand functions (2.1) into a Taylor series in a neighborhood of the point . Will have

Because the left-hand sides of (2.2) must vanish according to (1.1), then the right-hand sides of (2.2) must also vanish. Therefore, from (2.2) we have

All partial derivatives in (2.3) must be calculated at the point .

(2.3) is a system of linear algebraic equations relative to unknowns This system can be solved by Cramer’s method if its main determinant is nonzero and the quantities can be found

Now we can refine the zero approximation by constructing the first approximation with the coordinates

those.
. (2.6)

Let us find out whether approximation (2.6) has been obtained with a sufficient degree of accuracy. To do this, let's check the condition

,
(2.7)

Where a predetermined small positive number (the accuracy with which system (1.1) must be solved). If condition (2.7) is satisfied, then we choose (2.6) as an approximate solution to system (1.1) and complete the calculations. If condition (2.7) is not satisfied, then we perform the following action. In system (2.3), instead of
let's take the updated values

, (2.8)

those. let's do it the following actions

. (2.9)

After this, system (2.3) will be a system of linear algebraic equations for the quantities Having determined these quantities, the next second approximation
to the solution of system (1.1) we find using the formulas

Now let's check condition (2.7)

If this condition is met, then we complete the calculations by taking the second approximation as an approximate solution to system (1.1)
. If this condition is not met, then we continue to construct the next approximation, taking in (2.3)
It is necessary to build approximations until the condition is not satisfied.

The working formulas of Newton's method for solving system (1.1) can be written in the form.

Compute sequence

Here
are the solution to the system

Let us formulate a calculation algorithm using formulas (2.11)-(2.13).

1. Let us choose a zero approximation belonging to region D.

2. In the system of linear algebraic equations (2.13) we set
,A .

3. Let’s solve system (2.13) and find the quantities
.

4. In formulas (2.12) we put
and calculate the components of the next approximation.

5. Let's check condition (2.7) for: (See the algorithm for calculating the maximum of several quantities.)

6. If this condition is met, then we complete the calculations by choosing the approximation as the approximate solution to system (1.1). If this condition is not met, then move on to step 7.

7. Let's put
for all .

8. Let’s carry out step 3, putting
.

Geometrically, this algorithm can be written as:

Algorithm. Calculation of the maximum of several quantities.

Example. Let's consider using Newton's method to solve a system of two equations.

Solve using Newton's method up to accuracy the following system nonlinear equations

, (2.14)

Here
. Let's choose the zero approximation
, belonging to domain D. Let us construct a system of linear algebraic equations (2.3). She will look like

(2.15)

Let's denote

Let us solve system (2.15) with respect to unknowns
, for example the Cramer method. We write Cramer's formulas in the form

(2.17)

where is the main determinant of the system (2.15)

(2.18)

and the auxiliary determinants of system (2.15) have the form

.

We substitute the found values ​​into (2.16) and find the components of the first approximation
to the solution of system (2.15).

Let's check the condition

, (2.19)

if this condition is met, then we complete the calculations by taking the first approximation as an approximate solution to system (2.15), i.e.
. If condition (2.19) is not satisfied, then we set
,
and we'll build new system linear algebraic equations (2.15). Having solved it, we find the second approximation
. Let's check it on . If this condition is satisfied, then we choose as an approximate solution to system (2.15)
. If the condition on is not satisfied, we set
,
and construct the following system (2.15) to find
etc.

Tasks

All tasks require:

Draw up a program for the numerical implementation of the method according to the proposed algorithm.

Get calculation results.

Check your results.

A system of two nonlinear equations is given.

1.
2.

3.
4.

5.
6.

7.
8.

9.
10.

11.
12.

13.
14.

15.
.

Chapter 3. Numerical methods for solving systems of linear algebraic equations (SLAEs).

Goal of the work. Introduction to some approximate methods for solving SLAEs and their numerical implementation on a PC.

Preliminary remarks. All methods for solving SLAEs are usually divided into two large groups. The first group includes methods that are commonly called accurate. These methods allow us to find for any system exact values unknowns after a finite number of arithmetic operations, each of which is performed exactly.

The second group includes all methods that are not accurate. They are called iterative, or numerical, or approximate. The exact solution, when using such methods, is obtained as a result of an endless process of approximations. An attractive feature of such methods is their self-correction and ease of implementation on a PC.

Let us consider some approximate methods for solving SLAEs and construct algorithms for their numerical implementation. We will obtain an approximate solution of the SLAE with an accuracy of , where is some very small positive number.

1. Iteration method.

Let the SLAE be given in the form

(1.1)

This system can be written in matrix form

, (1.2)

Where
- matrix of coefficients for unknowns in system (1.1),
- column of free members,
- column of unknowns of the system (1.1).

. (1.3)

Let us solve system (1.1) using the iteration method. To do this, we will perform the following steps.

Firstly. Let's choose the zero approximation

(1.4)

to the exact solution (1.3) of system (1.1). The components of the zero approximation can be any numbers. But it is more convenient to take either zeros for the components of the zero approximation
, or free terms of system (1.1)

Secondly. We substitute the components of the zero approximation into right side system (1.1) and calculate

(1.5)

The quantities on the left in (1.5) are components of the first approximation
The actions that resulted in the first approximation are called iteration.

Third. Let's check the zero and first approximations for

(1.6)

If all conditions (1.6) are met, then for the approximate solution of system (1.1) we choose either , or it doesn’t matter, because they differ from each other no more than by and let’s finish the calculations. If at least one of the conditions (1.6) is not met, then we move on to the next action.

Fourthly. Let's perform the next iteration, i.e. into the right side of system (1.1) we substitute the components of the first approximation and calculate the components of the second approximation
, Where

Fifthly. Let's check
and on , i.e. Let us check condition (1.6) for these approximations. If all conditions (1.6) are met, then for the approximate solution of system (1.1) we will choose either , or it doesn’t matter, because they differ from each other by no more than . Otherwise, we will construct the next iteration by substituting the components of the second approximation into the right side of system (1.1).

Iterations need to be built until two adjacent approximations
and will differ from each other by no more than .

The working formula of the iteration method for solving system (1.1) can be written as

The algorithm for the numerical implementation of formula (1.7) can be as follows.

Sufficient conditions for the convergence of the iteration method for system (1.1) have the form

1.
, .

2.
,
.

3.

2. Simple iteration method.

Let the system of linear algebraic equations (SLAE) be given in the form

(2.1)

In order to solve system (2.1) using the simple iteration method, it must first be reduced to the form

(2.2)

In system (2.2) The -th equation is the -th equation of system (2.1), resolved with respect to the -th unknown (
).

The method for solving system (2.1), which consists of reducing it to system (2.2) followed by solving system (2.2) using the iteration method, is called the simple iteration method for system (2.1).

Thus, the working formulas of the simple iteration method for solving system (2.1) will have the form

(2.3)

Formulas (2.3) can be written in the form

The algorithm for the numerical implementation of the simple iteration method for system (2.1) according to formulas (2.4) can be as follows.

This algorithm can be written geometrically.

Sufficient conditions for the convergence of the simple iteration method for system (2.1) have the form

1.
, .

2.
,
.

3.

3. Stationary Seidel method.

The Seidel method for solving SLAEs differs from the iteration method in that having found some approximation for the -th component, we immediately use it to find the next
,
, …, -th component. This approach allows for more high speed convergence of the Seidel method compared to the iteration method.

Let the SLAE be given in the form

(3.1)

Let
- zero approximation to the exact solution
systems (3.1). And let it be found th approximation
. Let's define the components
th approximation using formulas

(3.2)

Formulas (3.2) can be written in compact form

,
,
(3.3)

The algorithm for the numerical implementation of the Seidel method for solving system (3.1) using formulas (3.3) can be as follows.

1. Let's choose, for example,
,

2. Let's put .

3. Let's calculate for everyone.

4. We will check the conditions for everyone
.

5. If all the conditions in paragraph 4 are met, then we will choose either or as an approximate solution to system (3.1) and complete the calculations. If at least one condition in step 4 is not met, move on to step 6.

6. Let’s put it down and move on to step 3.

This algorithm can be written geometrically.

The sufficient condition for the convergence of the Seidel method for system (3.1) has the form
, .

4. Non-stationary Seidel method.

This method of solving SLAE (3.1) provides an even higher speed of convergence of the Seidel method.

Let us somehow find the components of the th approximation and the th approximation for system (3.1).

Let's calculate the correction vector

Let's calculate the values

, (4.2)

Let's arrange the quantities
, in descending order.

In the same order, we rewrite the equations in system (3.1) and the unknowns in this system: Linearalgebra And nonlinear ... ManagementFor laboratory worksBy ... methodological instructions ForpracticalworksBy Forstudents ...

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Keywords:

Goal of the work: study methods for solving nonlinear equations with one unknown and test them in experimental work.

Job objectives:

1. Analyze special literature and choose the most rational methods for solving nonlinear equations, allowing you to deeply study and assimilate this topic all high school graduates.
2. Develop some aspects of a methodology for solving nonlinear equations using ICT.
3. Explore methods for solving nonlinear equations:

‒ Step method

‒ Halving method

‒ Newton's method

Introduction.

Without mathematical literacy, it is impossible to successfully master methods for solving problems in physics, chemistry, biology and other subjects. The entire complex of natural sciences is built and developed on the basis of mathematical knowledge. For example, the study of a number of topical problems in mathematical physics leads to the need to solve nonlinear equations. The solution of nonlinear equations is necessary in nonlinear optics, plasma physics, superconductivity theory, and low-temperature physics. There is a sufficient amount of literature on this topic, but many textbooks and articles are difficult for a high school student to understand. This paper discusses methods for solving nonlinear equations that can be used to solve applied problems in physics and chemistry. An interesting aspect is the application information technologies to solving equations and problems in mathematics.

Step method.

Let it be necessary to solve a nonlinear equation of the form F(x)=0. Let's also assume that we are given a certain search interval. It is required to find the interval [a,b] of length h, containing the first root of the equation, starting from the left border of the search interval.

Rice. 1. Step method

There are several ways to solve such a problem. The step method is the simplest of the numerical methods for solving inequalities, but to achieve high accuracy it is necessary to significantly reduce the step, and this greatly increases the calculation time. Algorithm for solving equations using this method consists of two stages.

Istage. Root separation.

At this stage, sections are determined, each of which contains only one root of the equation. There are several options for implementing this stage:

• We substitute the values ​​of X (preferably with some fairly small step) and see where the function changes sign. If the function has changed its sign, this means that there is a root in the area between the previous and current value of X (if the function does not change the nature of its increase/decrease, then we can say that there is only one root in this interval).
• Graphic method. We build a graph and evaluate on which intervals one root lies.
• Let's explore the properties of a specific function.

IIstage. Refinement of roots.

At this stage, the meaning of the roots of the equation determined earlier is clarified. As a rule, iterative methods are used at this stage. For example, the method half division(dichotomies) or Newton's method.

Half division method

A fast and fairly simple numerical method for solving equations, based on the sequential narrowing of the interval containing the only root of the equation F(x) = 0 until the specified accuracy E is achieved. This method is usually used when solving quadratic equations and equations of higher degrees. However, this method has a significant drawback - if the segment [a,b] contains more than one root, then it will not be able to achieve good results.

Rice. 2. Dichotomy method

The algorithm for this method is as follows:

‒ Determine a new approximation of the root x in the middle of the segment [a;b]: x=(a+b)/2.

‒ Find the values ​​of the function at points a and x: F(a) and F(x).

‒ Check the condition F(a)*F(x)

‒ Go to step 1 and again divide the segment in half. Continue the algorithm until the condition |F(x)|

Newton's method

The most accurate of the numerical solution methods; suitable for solving very complex equations, but is complicated by the need to calculate derivatives at each step. is that if x n is some approximation to the root of the equation , then the next approximation is defined as the root of the tangent to the function f(x) drawn at the point x n.

The tangent equation to the function f(x) at point x n has the form:

In the tangent equation we put y = 0 and x = x n +1.

Then the algorithm for sequential calculations in Newton’s method is as follows:

The convergence of the tangent method is quadratic, the order of convergence is 2.

Thus, the convergence of Newton's tangent method is very fast.

Without any changes, the method is generalized to the complex case. If the root x i is a root of the second multiplicity or higher, then the order of convergence drops and becomes linear.

The disadvantages of Newton's method include its locality, since it is guaranteed to converge for an arbitrary starting approximation only if the condition is satisfied everywhere , in the opposite situation, convergence occurs only in a certain neighborhood of the root.

Newton's method (tangent method) is usually used when the equation f(x) = 0 has a root and the following conditions are met:

1) function y=f(x) defined and continuous at ;

2) f(a) f(b) (the function takes values ​​of different signs at the ends of the segment [ a;b]);

3) derivatives f"(x) And f""(x) preserve sign on the interval [ a;b] (i.e. the function f(x) either increases or decreases on the segment [ a;b], while maintaining the direction of the convexity);

The meaning of the method is as follows: on the segment [ a;b] such a number is selected x 0 , at which f(x 0) has the same sign as f""(x 0), i.e. the condition is satisfied f(x 0) f""(x) > 0. Thus, the point with the abscissa is selected x 0, in which the tangent to the curve y=f(x) on the segment [ a;b] intersects the axis Ox. Per point x 0 First, it is convenient to select one of the ends of the segment.

Let's consider this algorithm using a specific example.

Let us be given an increasing function y = f(x) =x 2– 2, continuous on the segment (0;2), and having f "(x) =2x>0 And f ""(x) = 2> 0.

In our case, the tangent equation has the form: y-y 0 =2x 0 ·(x-x 0). IN as point x 0 we select point B 1 (b; f(b)) = (2,2). Draw a tangent to the function y = f(x) at point B 1, and denote the point of intersection of the tangent and axis Ox dot x 1. We get the equation of the first tangent: y-2=2·2(x-2), y=4x-6. Ox: x 1 =

Rice. 3. Construction of the first tangent to the graph of the function f(x)

y=f(x) Ox through the point x 1, we get the point B 2 =(1.5; 0.25). Draw a tangent to the function again y = f(x) at point B 2, and denote the point of intersection of the tangent and Ox dot x 2.

Equation of the second tangent: y-2.25=2*1.5(x-1.5), y = 3x - 4.25. Intersection point of tangent and axis Ox: x 2 =.

Then we find the intersection point of the function y=f(x) and a perpendicular drawn to the axis Ox through point x 2, we get point B 3 and so on.

Rice. 4. Construction of the second tangent to the graph of the function f(x)

The first approximation of the root is determined by the formula:

= 1.5.

The second approximation of the root is determined by the formula:

=

The third approximation of the root is determined by the formula:

Thus ,i The th approximation of the root is determined by the formula:

Calculations are carried out until the decimal places that are needed in the answer match, or the specified precision e is achieved - until the inequality is satisfied |xi-xi-1|

In our case, let's compare the approximation obtained in the third step with the real answer. As you can see, already at the third step we received an error of less than 0.000002.

Solving an equation using CADMathCAD

For the simplest equations of the form f(x) = 0 the solution in MathСAD is found using the function root.

root(f (X 1 , x 2 , … ) , X 1 , a, b ) - returns value X 1 , belonging to the segment [ a, b ] , in which the expression or function f (X ) goes to 0. Both arguments to this function must be scalars. The function returns a scalar.

Rice. 5. Solving a nonlinear equation in MathCAD (root function)

If an error occurs as a result of applying this function, this may mean that the equation has no roots, or the roots of the equation are located far from the initial approximation, the expression has local max And min between the initial approximation and the roots.

To establish the cause of the error, it is necessary to examine the graph of the function f(x). It will help to find out the presence of roots of the equation f(x) = 0 and, if they exist, then approximately determine their values. The more accurately the initial approximation of the root is chosen, the faster its exact value will be found.

If the initial approximation is unknown, then it is advisable to use the function solve . Moreover, if the equation contains several variables, you need to indicate after keyword solve is a list of variables for which the equation is solved.

Rice. 6. Solving a nonlinear equation in MathCAD (solve function)

Conclusion

The study examined how mathematical methods, and solving equations using programming in the CAD system MathCAD. Various methods have their advantages and disadvantages. It should be noted that the use of a particular method depends on the initial conditions of the given equation. Those equations that can be solved well by methods of factorization, etc., known in school, do not make sense to solve more in complex ways. Applied mathematics problems that are important for physics and chemistry and require complex computational operations when solving equations are successfully solved, for example, using programming. It’s good to solve them using Newton’s method.

To clarify the roots, you can use several methods for solving the same equation. It was this research that formed the basis of this work. At the same time, it is easy to see which method is most successful when solving each stage of the equation, and which method is better not to use at this stage.

The studied material, on the one hand, helps to expand and deepen mathematical knowledge and instill interest in mathematics. On the other hand, it is important to be able to solve real mathematics problems for those who are planning to acquire technical and engineering professions. That's why this work matters for further education(for example, in a higher education institution).

Literature:

1. Mityakov S. N. Informatics. Complex educational materials. - N. Novgorod: Nizhny Novgorod. state tech. univ., 2006
2. Vainberg M. M., Trenogin V. A. The theory of branching solutions of nonlinear equations. M.: Nauka, 1969. - 527 p.
3. Bronshtein I. N., Semendyaev K. A. Handbook of mathematics for engineers and students of technical colleges - M.: Nauka, 1986.
4. Omelchenko V. P., Kurbatova E. V. Mathematics: tutorial. - Rostov n/d.: Phoenix, 2005.
5. Savin A.P. encyclopedic Dictionary young mathematician. - M.: Pedagogy, 1989.
6. Korn G., Korn T. Handbook of mathematics for scientists and engineers. - M.: Nauka, 1973.
7. Kiryanov D. Mathcad 15/MathcadPrime 1.0. - St. Petersburg: BHV-Petersburg, 2012.
8. Chernyak A., Chernyak Zh., Domanova Yu. Higher mathematics based on Mathcad. General course. - St. Petersburg: BHV-Petersburg, 2004.
9. Porshnev S., Belenkova I. Numerical methods based on Mathcad. - St. Petersburg: BHV-Petersburg, 2012.

Keywords: nonlinear equations, applied mathematics, CAD MathCAD, Newton's method, step method, dichotomy method..

Annotation: The article is devoted to the study of methods for solving nonlinear equations, including using the MathCAD computer-aided design system. The step method, halves and Newton methods are considered, detailed algorithms for applying these methods are given, and comparative analysis the specified methods.

For example:

Let's set the task to find valid roots of this equation.

And there definitely are! - from articles about function graphs And equations of higher mathematics you know very well what the schedule is polynomial function odd degree intersects the axis at least once, therefore our equation has at least one real root. One. Or two. Or three.

First, it begs to check the availability rational roots. According to corresponding theorem, only the numbers 1, –1, 3, –3 can claim this “title”, and by direct substitution it is easy to make sure that none of them “suits”. Thus, irrational values ​​remain. The irrational root(s) of a polynomial of degree 3 can be found exactly (express through radicals) using the so-called Cardano formulas , however, this method is quite cumbersome. But for polynomials of the 5th and higher degrees there is no general analytical method at all, and, in addition, in practice there are many other equations in which exact values it is impossible to obtain real roots (although they exist).

However, in applied (for example, engineering) problems, it is more than acceptable to use approximate values ​​calculated with a certain accuracy.

Let's set the accuracy for our example. What does it mean? This means that we need to find SUCH an approximate value of the root (roots) in which we we are guaranteed to be wrong by no more than 0.001 (one thousandth) .

It is absolutely clear that the solution cannot be started “at random” and therefore in the first step the roots separate. To separate a root means to find a sufficiently small (usually single) segment to which this root belongs and on which there are no other roots. The simplest and most accessible graphical method of root separation. Let's build point by point graph of a function :

From the drawing it follows that the equation, apparently, has a single real root belonging to the segment. At the ends of this interval the function takes values ​​of different signs: , and from the fact continuity of the function on the segment immediately visible elementary way root refinement: divide the interval in half and select the segment at the ends of which the function takes different signs. IN in this case this is obviously a segment. We divide the resulting interval in half and again select the “different sign” segment. And so on. Such sequential actions are called iterations. In this case, they should be carried out until the length of the segment becomes less than twice the calculation accuracy, and the middle of the last “different-sign” segment should be chosen as the approximate value of the root.

The considered scheme received a natural name - half division method. And the disadvantage of this method is speed. Slowly. So slow. There will be too many iterations before we achieve the required accuracy. With development computer technology This, of course, is not a problem, but that’s what mathematics is for, to look for the most rational solutions.

And one of the more effective ways finding the approximate value of the root is precisely tangent method. The brief geometric essence of the method is as follows: first, using a special criterion (more on that a little later) one of the ends of the segment is selected. This end is called initial approximation of the root, in our example: . Now we draw a tangent to the graph of the function at the abscissa point (blue dot and purple tangent):

This tangent crossed the x-axis at the yellow point, and note that in the first step we have almost “hit the root”! It will be first root approach. Next, we lower the yellow perpendicular to the graph of the function and “get” to the orange point. We again draw a tangent through the orange point, which will intersect the axis even closer to the root! And so on. It is not difficult to understand that using the tangent method, we are approaching the goal by leaps and bounds, and it will take literally several iterations to achieve accuracy.

Since the tangent is defined through derivative of the function, then this lesson ended up in the “Derivatives” section as one of its applications. And without going into detail theoretical justification of the method, I will consider the technical side of the issue. In practice, the problem described above occurs approximately in the following formulation:

Example 1

By using graphic method find the interval on which the real root of the equation is located. Using Newton's method, obtain an approximate value of the root with an accuracy of 0.001

Here is a “sparing version” of the task, in which the presence of a single valid root is immediately stated.

Solution: on the first step the root should be separated graphically. This can be done by plotting (see illustrations above), but this approach has a number of disadvantages. Firstly, it’s not a fact that the graph is simple (we don’t know in advance), A software– it is not always at hand. And secondly (corollary from 1st), with considerable probability the result will not even be a schematic drawing, but a rough drawing, which, of course, is not good.

Well, why do we need unnecessary difficulties? Let's imagine the equation in the form, CAREFULLY construct graphs and mark the root in the drawing (“X” coordinate of the point of intersection of the graphs):

Obvious advantage this method is that graphs of these functions are built by hand much more accurately and much faster. By the way, note that straight crossed cubic parabola at a single point, which means that the proposed equation actually has only one real root. Trust, but verify ;-)

So, our “client” belongs to the segment and “by eye” is approximately equal to 0.65-0.7.

On the second step need to choose initial approximation root Usually this is one of the ends of the segment. The initial approximation must satisfy next condition:

Let's find first And second derived functions :

and check the left end of the segment:

Thus, the zero “did not fit.”

Checking the right end of the segment:

- Everything is fine! We choose as the initial approximation.

On the third step The road to the root awaits us. Each subsequent root approximation is calculated from the previous data using the following recurrent formulas:

The process ends when the condition is met, where is the predetermined accuracy of calculations. As a result, the “nth” approximation is taken as the approximate value of the root: .

Next up are the routine calculations:

(rounding is usually carried out to 5-6 decimal places)

Since the obtained value is greater than , we proceed to the 1st approximation of the root:

We calculate:

, so there is a need to move to the 2nd approximation:

Let's move on to the next round:

, thus, the iterations are completed, and the 2nd approximation should be taken as the approximate value of the root, which, in accordance with the given accuracy, should be rounded to one thousandth:

In practice, it is convenient to enter the results of calculations into a table; in order to shorten the entry somewhat, a fraction is often denoted by:

If possible, it is better to carry out the calculations themselves in Excel - it is much more convenient and faster:

Answer: accurate to 0.001

Let me remind you that this phrase implies the fact that we made a mistake in our assessment true meaning root by no more than 0.001. Those in doubt can pick up a microcalculator and once again substitute the approximate value of 0.674 in left side equations

Now let’s “scan” the right column of the table from top to bottom and notice that the values ​​are steadily decreasing in absolute value. This effect is called convergence a method that allows us to calculate the root with arbitrarily high accuracy. But convergence does not always take place - it is ensured a number of conditions, about which I kept silent. In particular, the segment on which the root is isolated must be small enough– otherwise the values ​​will change randomly and we will not be able to complete the algorithm.

What to do in such cases? Check that the specified conditions are met (see link above), and, if necessary, reduce the segment. So, relatively speaking, if in the analyzed example the interval was not suitable for us, then we should consider, for example, the segment. In practice, I have encountered such cases, and this technique really helps! The same must be done if both ends of the “wide” segment do not satisfy the condition (i.e., none of them are suitable as an initial approximation).

But usually everything works like a clock, although not without pitfalls:

Example 2

Determine graphically the number of real roots of the equation, separate these roots and, using Newton’s method, find approximate values ​​of the roots with accuracy

The condition of the problem has become noticeably stricter: firstly, it contains a strong hint that the equation does not have a single root, secondly, the requirement for accuracy has increased, and thirdly, with the graph of the function much more difficult to cope with.

And therefore solution Let's start with a saving trick: imagine the equation in the form and draw graphs:


From the drawing it follows that our equation has two real roots:

The algorithm, as you understand, needs to be “cranked” twice. But this is even in the most severe cases; sometimes you have to examine 3-4 roots.

1) Using criterion Let's find out which end of the segment to choose as the initial approximation of the first root. Finding derivatives of functions :

Testing the left end of the segment:

- came up!

Thus, is an initial approximation.

We will refine the root using Newton’s method using the recurrent formula:
- until the fraction modulo will not be less than the required accuracy:

And here the word “module” acquires non-illusory importance, since the values ​​​​are negative:


For the same reason, you should show increased attention when moving to each next approximation:

Despite enough high requirement to accuracy, the process ended again at the 2nd approximation: , therefore:

Accurate to 0.0001

2) Let's find the approximate value of the root.

We check the left end of the segment for lice:

, therefore, it is not suitable as an initial approximation.