高等應(yīng)用數(shù)學(xué)問題的matlab求解06

1、星期四, 2008-4- 24, 22:08:13,Slide 1 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Chapter 6Nonlinear Equations and

2、 Optimization Problems,Solving Applied Mathematical Problems with MATLAB,CRC/Taylor & Francis PressChinese version by Tsinghua University Press,PPT by Wenbin Dong and Jun Peng, Northeastern University, PRCProofread

3、 by Dingyu Xue & YangQuan Chen,星期四, 2008-4- 24, 22:08:13,Slide 2 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 200

4、8,Chapter 6 Nonlinear Equations and Optimization Problems,Solutions to Nonlinear Algebraic EquationsSolving Unconstrained Optimization ProblemsSolving Constrained Optimization ProblemsSolving Mixed Integer Programmin

5、g ProblemsLinear Matrix Inequalities,星期四, 2008-4- 24, 22:08:13,Slide 3 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press,

6、2008,6.1 Solutions to Nonlinear Algebraic Equations,Graphical method for solving nonlinear equationsQuasi-analytical solutions to polynomial-type equationsNumerical solutions to ordinary nonlinear equationsSolutions t

7、o nonlinear matrix equations,星期四, 2008-4- 24, 22:08:13,Slide 4 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,6.1.

8、1 Graphical method for solving nonlinear equations,Graphically solving nonlinear equations of a single variableGraphically solving nonlinear equations of two variablesNot suitable more equations with more variables,星期四

9、, 2008-4- 24, 22:08:13,Slide 5 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Graphically solving nonlinear equati

10、ons of a single variable,The function ezplot() can be used to draw the curves from the implicit function . The real solutions to the equations can be retrieved from the intersections of the cu

11、rves with,星期四, 2008-4- 24, 22:08:13,Slide 6 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 6.1,Graphically

12、 solve :MATLAB commands:Verification:,星期四, 2008-4- 24, 22:08:13,Slide 7 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC

13、Press, 2008,Graphically solving nonlinear equations of two variables,Use ezplot() function to draw the all the equations. The intersections of the curves are then the solutions to the original simultaneous equations,星期四

14、, 2008-4- 24, 22:08:13,Slide 8 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 6.2,Graphically solve the fo

15、llowing:Plot of the first function:Plot of the second function:,星期四, 2008-4- 24, 22:08:13,Slide 9 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical

16、Problems with MATLAB, CRC Press, 2008,The rectangular region can be selected,星期四, 2008-4- 24, 22:08:13,Slide 10 (of 138) Dingyü Xue and YangQuan Chen, Solving App

17、lied Mathematical Problems with MATLAB, CRC Press, 2008,6.1.2 Quasi-analytical solutions to polynomial-type equations,Special equations such as polynomial equations, may be of high orderAbel-Ruffini Theorem

18、 claim that no analytical solution method for 5+th order equationNumerical solution not accurateHigh precision methods existMany equations can be converted into polynomial equations,星期四, 2008-4- 24, 22:08:13,Slide

19、11 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 6.3,Graphically solveMATLAB command solutions:,星期四, 20

20、08-4- 24, 22:08:13,Slide 12 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,The syntax to solve polynomial-type equa

21、tionsthe simplestdirect solutionsvariables specified,星期四, 2008-4- 24, 22:08:13,Slide 13 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems w

22、ith MATLAB, CRC Press, 2008,Example 6.4,Applying solve() function to solveMATLAB command solutions:Verification:,星期四, 2008-4- 24, 22:08:13,Slide 14 (of 138) Dingyü Xue and YangQuan

23、 Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 6.5,Solve MATLAB command solutions:Verification:,星期四, 2008-4- 24, 22:08:13,Slide 15 (of 138)

24、 Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,When the last equation is changed intoMATLAB command solutions:,星期四, 2008-4- 24, 22:08:13,Slide

25、 16 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 6.6,SolveMATLAB command solutions:,星期四, 2008-4- 24,

26、 22:08:13,Slide 17 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Verification:,星期四, 2008-4- 24, 22:08:13,Slide 1

27、8 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 6.7,Solve the equations with constantsMATLAB command sol

28、utions:,星期四, 2008-4- 24, 22:08:13,Slide 19 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,6.1.3 Numerical solution

29、s to ordinary nonlinear equations,The syntax of getting real numerical solutions to given nonlinear equationssimple syntaxformal syntax,星期四, 2008-4- 24, 22:08:13,Slide 20 (of 138) Dingy

30、52; Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,The syntax of selecting methods and control accuraciesget default control templateset control parametersO

31、r,星期四, 2008-4- 24, 22:08:13,Slide 21 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Procedures in Solving Numerica

32、lly Algebraic Equations,Set the variables, so that the equations areDescribe the equations withM-functionAnonymous functionInline function, again not recommendedSolve the equations Verify the solutions,星期四, 2008-

33、4- 24, 22:08:13,Slide 22 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 6.8,Given select variables and

34、 convert the original ODEs toMatrix form,星期四, 2008-4- 24, 22:08:13,Slide 23 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC P

35、ress, 2008,Ways of describing the equation,M-functionAnonymous functionInline function,星期四, 2008-4- 24, 22:08:13,Slide 24 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied

36、 Mathematical Problems with MATLAB, CRC Press, 2008,When the initial values areWhen another initial search point usedNOTE: the selection of different initial values may lead to different solutions.,星期四,

37、 2008-4- 24, 22:08:13,Slide 25 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 6.9,Lambert function

38、 , is variable, is the solution of . For different , solve and plot .Solution tactics and procedureUse for loopUse anonymous function to describeGenerate w vectorPlo

39、t the curve of the functionMATLAB solutions:,星期四, 2008-4- 24, 22:08:13,Slide 26 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CR

40、C Press, 2008,MATLAB statementsDirect solution, with lambertw function:,星期四, 2008-4- 24, 22:08:13,Slide 27 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Math

41、ematical Problems with MATLAB, CRC Press, 2008,Example 6.10,Solve numericallyUsing the solve() functionUsing graphical method for initial value:,星期四, 2008-4- 24, 22:08:13,Slide 28 (of 138)

42、 Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Re-setting the relevant precision control variablesThe expected precision may not be reachedHowe

43、ver the best result under double-precision scheme can be obtained,星期四, 2008-4- 24, 22:08:13,Slide 29 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Prob

44、lems with MATLAB, CRC Press, 2008,Summary of three types of algebraic equation solution methods,Graphical methods, ezplotAdvantages: all real solutions within a region can be found; good overview for the solutionsDisad

45、vantages: Only applicable to 1D and 2D equations; not accurate; only real solutions Quasi-analytical methods, solveAdvantages: Real/imaginary, more accurateDisadvantages: Only applies to polynomial type equations; not

46、 allowed to select initial point,星期四, 2008-4- 24, 22:08:13,Slide 30 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

47、,Pure numerical methods fsolveAdvantages: Can be used to solve multi-dimensional nonlinear equations; freely select initial search point, accurate than graphical methodDisadvantages: Imaginary solutions cannot be foun

48、d; for polynomial type equations, not so goodCombination of algorithms, hintsFor 1D and 2D equations, use graphical method to spot an approximate solutionUse the solution as an initial point, search for more accurate

49、solutions with fsolve,星期四, 2008-4- 24, 22:08:13,Slide 31 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,6.1.4 Solu

50、tions to nonlinear matrix equations,Riccati equation in Chapter 4, explorationMore nonlinear matrix equations, e.g.,Generalized Riccati equationModified Riccati equationMany many more matrix equations,星期四, 2008-

51、4- 24, 22:08:13,Slide 32 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,fsolve() can only solve , not

52、 vector , vector , no matrices involvedConvert matrix equation into vector equationsVector to matrix math MATLABMatrix to vector mathMATLAB,Riccati equation solver,星期四, 2008-4- 24, 22:08:13,Slide 33 (

53、of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,M-function to describe Riccati equation in vector formA new function

54、 to solve Riccati equation,星期四, 2008-4- 24, 22:08:13,Slide 34 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Examp

55、le 6.11,Solve the following Riccati equations: with,星期四, 2008-4- 24, 22:08:13,Slide 35 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with

56、MATLAB, CRC Press, 2008,are() function may find oneRepeated use of MATLAB commands:Another solution,星期四, 2008-4- 24, 22:08:13,Slide 36 (of 138) Dingyü Xue and YangQuan Chen, Solvin

57、g Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 6.12,Givenwhere find and verify all the possible solutions,星期四, 2008-4- 24, 22:08:13,Slide 37 (of 138)

58、 Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,For the Riccati-like equationAnother M-functionAnother matrix equation solver,星期四, 2008-4- 24,

59、22:08:13,Slide 38 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Repeated MATLAB command:Be patient, some of

60、the solutions may be difficult to find. Run the function multiple times,星期四, 2008-4- 24, 22:08:13,Slide 39 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematica

61、l Problems with MATLAB, CRC Press, 2008,Possible solutions, all passed verifications,星期四, 2008-4- 24, 22:08:13,Slide 40 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied

62、 Mathematical Problems with MATLAB, CRC Press, 2008,6.2 Solving Unconstrained Optimization Problems,Analytical solutions and graphical solution methodsNumerical solution using MATLABGlobal minima and local minimaSolv

63、ing optimization problems using gradient informationOptimization problems with boundary constraints,星期四, 2008-4- 24, 22:08:13,Slide 41 (of 138) Dingyü Xue and YangQuan Chen, Solving App

64、lied Mathematical Problems with MATLAB, CRC Press, 2008,The mathematical description to unconstrained minimization problemsObjective function is a scalar onevector

65、 decision variables, or optimum variables Physical meaning: find the vector that minimizes the objective functionMaximization problem,,Mathematical description,星期四, 2008-4- 24, 22:08:13,Slide 42 (of 138)

66、 Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,6.2.1 Analytical solutions and graphical solution methods,The necessary conditions for

67、 an unconstrained optimization problemwhere is the optimum pointEquation solution may be more difficult, also second-order derivatives needed,星期四, 2008-4- 24, 22:08:13,Slide 43 (of 138)

68、 Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 6.13,Study the optimality of Plot of the first order derivative function of,星期四, 2008-4-

69、 24, 22:08:13,Slide 44 (of 138) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Draw the first-order derivative and verify po