Numerical Methods – 5. Iterative methods for nonlinear equations

Numerical Methods

Handouts for students

5. Iterative methods for nonlinear equations

5.1Bounding the solution

5.2Bisection Method

5.3Iteration Method

5.4Tangent Method (Newton’s Method)

5.5Secant Method (RegulaFalsi Method)

5.6Newton’s Method for systems of nonlinear equations

  1. Introductoryrequirements

It is required to know the concepts of:

  • secant, tangent line;
  • derivative offunction of one variable;
  • characterisation of properties of function via derivatives;
  • jacobian;
  • inversematrix;

and be able to:

  • solvelinearequations;
  • given two points, find the formula of the line;
  • compute first and second derivatives of function of one variable and multiple variables;
  • find tangent line to the graph of function.
  1. Classes

Task 1.Bound the roots of given equations to the intervals of length 1:

a);

b);

c).

Task 2.For the equations and intervals from the task 1, find the second approximation of each solution using the bisection method. Evaluate the absolute errors of these approximations. How many iterations of the method is needed to ensure that the error of the approximation in no greater than

Task 3.Determineif the mapping satisfies the assumptions ofthe theorem on iteration method convergence:

a)on theinterval;

b)on theinterval.

Task 4.Determine which of the following reformulations of the equation guarantees the convergence of the Iteration Method. Use it to find the second approximation of the solution:

a)

b)

c)

d).

For each possible case, evaluate the absolute error of the second approximation (use the appropriate theorem from the lecture).

Task 5.Evaluate the first two iterations in the Newton’s Method for the polynomial in the interval . Prove that there exists exactly one zero of that polynomial in this interval. Justify the convergence of the method and calculate the absolute error of the second approximation of the solution on the basis of the appropriate theorem from the lecture.

Task 6.Using the RegulaFalsi find the first two approximations of the positive root of the equation

Justify the convergence of the method and calculate the absolute error of the second approximation of the solution on the basis of the appropriate theorem from the lecture.

Task 7.Find the third approximation of the numberusing:

a)the Bisection Method (starting from the interval of the length 1);

b)the Iteration Method;

c)theSecantMethod;

d)the Tangent Method.

Task 8.The functions of supply and demand for bananas are expressed by formulas and respectively, where represents the price of bananas. Using the Iteration Method calculate economic equilibrium price for bananas, taking as an answer the second iteration of this method.

Task 9.The dependence between the specific heatof waterat a temperature, and thetemperatureis described by formula Calculate thetemperature such that thespecific heat of waterreaches thegreatest value and calculate this value. Take as an answer thefirst iterationof theNewton’s Method.

Task 10.The section modulus of a rectangular beam, lying horizontally, is given by formula , whereis the width, – the height of the beam section. How to cuta cylindrically shaped stem,whose basehas a diameterequal 2,to obtain a rectangularbeamwith the highestsection modulus(seeFig.1)?As an answer take the second iteration of the Secant Method.

Task 11.Using the Newton’s Method evaluate the given number of iterations:

a) starting point, one iteration

b) starting point, two iterations

Task 12.Jaś Kowalskiis anappliedcomputer sciencestudentat the Cracow Universityof Economics.He wenton vacation toher grandparentsin France who arefamousmathematicians. Since he has not seen his grandparents for a long time, he asked themhow old they were. The old menanswered himas follows: “Think about the numbers describing how old we are. The difference of the squaresof these numbersis 240, and the differenceof cubes of these numbers is 21602". Jaś was alsoaskedhow old he thinks his grandparents are.Jaśgave themthis answer: “You are very vital and vivacious. I suppose you are 60 years oldbut taking intoaccount thesocial stereotypeconnected with marriagethat a man shouldbeelder than awomen, I guessthat grandmais58 years old, and grandpa is 60 years old". Grandparepliedto this as follows: “You're right, but this is notthe exactanswer". Grandmotherasked him: 'Do you know how to calculate how old we are?Or do you need aclue? ". Jaśquicklyand proudlyreplied: “I do not need a clue, I’m going to use Newton's Methodfor nonlinearequations.I learnedthis method duringthe courseofnumerical methodsat the Cracow Universityof Economics". What isthe correct answer that Jaśshouldgive?

  1. Homework

Task 1.For thegiven equations:

  1. localizeat least one of the solutions and if this is possible, indicate the fixed end of the interval;
  2. find the first four aproximations of the solution in the BisectionMethod (starting from the interval of the length 1);
  3. find the first four aproximations of the solution in the Iteration Method;
  4. find the first four aproximationsof the solution in the Falsi Method;
  5. find the first four aproximationsof the solution in the Newton’s Method.

a);

b);

c).

Task 2.For the equation:

a)derive the formula for the -st approximation in the Newton’sMethod;

b)taking, calculate.

Task 3.Evaluate the fifth approximation ofin:

a)theiterationmethod;

b)thesecantmethod;

c)thetangentmethod.

Task 4.Atoy airplane, powered by batteries, thrown upward in the air at an angle to the horizontal, moves along path described by formula .Find the maximum altitude of the airplane. As an answer take the second iteration of the Newton’s Method.

Task 5.Find the first two approximations of the solutions to the given systems of nonlinear equations using the Newton’sMethod:

a), ;

b), .

  1. Answers

Task 1.

a)

  1. , is the fixed end
  2. ; , ; ;
  3. ;

; , ; ;

  1. ; ; ;
    ;
  2. ; ; ;
    ;

b)

  1. , is the fixed end
  2. ; , ; ;
  3. ;

; ; ; ;
;

  1. ; ; ;
    ;
  2. ; ; ;
    .

c)

  1. , is the fixed end
  2. ; , ; ;
  3. ;

; ; ; ;

;

  1. ; ; ;

;

  1. ; ; ;

;

Task 2.

a)

b)

Task 3.

a) (, )

b) ()

c) ()

Task 4.. By graphical method we see that this equation has exactly one solution belonging to the interval. Using the Newton’s Method we get:
, Since for, then at functionreaches its maximum, which approximately equals 2,76949666.

Task 5.

a), ;

b).

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