求方程f(x)=x3-sinx-12x+1的全部实根,ε=10-6.用MATLAB编程,

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求方程f(x)=x3-sinx-12x+1的全部实根,ε=10-6.用MATLAB编程,
求方程f(x)=x3-sinx-12x+1的全部实根,ε=10-6.
用MATLAB编程,

用导数知识分析其有根区间为：（-4,-2）,（-2,2）,（2,6）.（分析略,可参看下图）

在matlab中保存为:bisection.m

function rtn=bisection(fx,xa,xb,n,delta)

% 二分法解方程

% fx是由方程转化的关于x的函数,有fx=0.

% xa 解区间上限

% xb 解区间下限

%解区间人为判断输入

% n 最多循环步数,防止死循环.

%delta 为允许误差

x=xa;fa=eval(fx);

x=xb;fb=eval(fx);

disp(' [ n xa xb xc fc ]');

for i=1:n

xc=(xa+xb)/2;x=xc;fc=eval(fx);

X=[i,xa,xb,xc,fc];

disp(X),

if fc*fa<0

xb=xc;

else xa=xc;

end

if (xb-xa)<delta,break,end

end

一、在区间（-4,-2）上的

>> y='x.^3-sin(x)-12*x+1';

>> bisection(y,-4,-2,1000,10^-6)

[ n xa xb xc fc ]

1.00000000000000 -4.00000000000000 -2.00000000000000 -3.00000000000000 10.14112000805987

2.00000000000000 -4.00000000000000 -3.00000000000000 -3.50000000000000 -0.22578322768962

3.00000000000000 -3.50000000000000 -3.00000000000000 -3.25000000000000 5.56367986546989

4.00000000000000 -3.50000000000000 -3.25000000000000 -3.37500000000000 2.82534681259798

5.00000000000000 -3.50000000000000 -3.37500000000000 -3.43750000000000 1.33949563431115

6.00000000000000 -3.50000000000000 -3.43750000000000 -3.46875000000000 0.56686165921837

7.00000000000000 -3.50000000000000 -3.46875000000000 -3.48437500000000 0.17305022015525

8.00000000000000 -3.50000000000000 -3.48437500000000 -3.49218750000000 -0.02573754654618

9.00000000000000 -3.49218750000000 -3.48437500000000 -3.48828125000000 0.07381342529514

10.00000000000000 -3.49218750000000 -3.48828125000000 -3.49023437500000 0.02407723034585

11.00000000000000 -3.49218750000000 -3.49023437500000 -3.49121093750000 -0.00082033300097

12.00000000000000 -3.49121093750000 -3.49023437500000 -3.49072265625000 0.01163090465268

13.00000000000000 -3.49121093750000 -3.49072265625000 -3.49096679687500 0.00540589985774

14.00000000000000 -3.49121093750000 -3.49096679687500 -3.49108886718750 0.00229293694096

15.00000000000000 -3.49121093750000 -3.49108886718750 -3.49114990234375 0.00073634034872

16.00000000000000 -3.49121093750000 -3.49114990234375 -3.49118041992188 -0.00004198673138

17.00000000000000 -3.49118041992188 -3.49114990234375 -3.49116516113281 0.00034717920735

18.00000000000000 -3.49118041992188 -3.49116516113281 -3.49117279052734 0.00015259683766

19.00000000000000 -3.49118041992188 -3.49117279052734 -3.49117660522461 0.00005530520306

20.00000000000000 -3.49118041992188 -3.49117660522461 -3.49117851257324 0.00000665927332

21.00000000000000 -3.49118041992188 -3.49117851257324 -3.49117946624756 -0.00001766371965

由结果可知：x1=-3.49117946624756 ；

二、在区间（-2,2）上的

>> bisection(y,-2,2,1000,10^-6)

[ n xa xb xc fc ]

1 -2 2 0 1

2.00000000000000 0 2.00000000000000 1.00000000000000 -10.84147098480790

3.00000000000000 0 1.00000000000000 0.50000000000000 -5.35442553860420

4.00000000000000 0 0.50000000000000 0.25000000000000 -2.23177895925452

5.00000000000000 0 0.25000000000000 0.12500000000000 -0.62272160838523

6.00000000000000 0 0.12500000000000 0.06250000000000 0.18778482278262

7.00000000000000 0.06250000000000 0.12500000000000 0.09375000000000 -0.21778875662614

8.00000000000000 0.06250000000000 0.09375000000000 0.07812500000000 -0.01506871423176

9.00000000000000 0.06250000000000 0.07812500000000 0.07031250000000 0.08634303568347

10.00000000000000 0.07031250000000 0.07812500000000 0.07421875000000 0.03563319753740

11.00000000000000 0.07421875000000 0.07812500000000 0.07617187500000 0.01028122478907

12.00000000000000 0.07617187500000 0.07812500000000 0.07714843750000 -0.00239400219556

13.00000000000000 0.07617187500000 0.07714843750000 0.07666015625000 0.00394354733549

14.00000000000000 0.07666015625000 0.07714843750000 0.07690429687500 0.00077475652874

15.00000000000000 0.07690429687500 0.07714843750000 0.07702636718750 -0.00080962685008

16.00000000000000 0.07690429687500 0.07702636718750 0.07696533203125 -0.00001743616404

17.00000000000000 0.07690429687500 0.07696533203125 0.07693481445313 0.00037865993160

18.00000000000000 0.07693481445313 0.07696533203125 0.07695007324219 0.00018061182108

19.00000000000000 0.07695007324219 0.07696533203125 0.07695770263672 0.00008158781284

20.00000000000000 0.07695770263672 0.07696533203125 0.07696151733398 0.00003207582048

21.00000000000000 0.07696151733398 0.07696533203125 0.07696342468262 0.00000731982724

22.00000000000000 0.07696342468262 0.07696533203125 0.07696437835693 -0.00000505816865

由结果可知：x2=0.07696437835693；

三、在区间（2,6）上的

>> bisection(y,2,6,1000,10^-6)

[ n xa xb xc fc ]

1.00000000000000 2.00000000000000 6.00000000000000 4.00000000000000 17.75680249530792

2.00000000000000 2.00000000000000 4.00000000000000 3.00000000000000 -8.14112000805987

3.00000000000000 3.00000000000000 4.00000000000000 3.50000000000000 2.22578322768962

4.00000000000000 3.00000000000000 3.50000000000000 3.25000000000000 -3.56367986546989

5.00000000000000 3.25000000000000 3.50000000000000 3.37500000000000 -0.82534681259798

6.00000000000000 3.37500000000000 3.50000000000000 3.43750000000000 0.66050436568885

7.00000000000000 3.37500000000000 3.43750000000000 3.40625000000000 -0.09227275798887

8.00000000000000 3.40625000000000 3.43750000000000 3.42187500000000 0.28164331500236

9.00000000000000 3.40625000000000 3.42187500000000 3.41406250000000 0.09406835756613

10.00000000000000 3.40625000000000 3.41406250000000 3.41015625000000 0.00074371965680

11.00000000000000 3.40625000000000 3.41015625000000 3.40820312500000 -0.04580302043948

12.00000000000000 3.40820312500000 3.41015625000000 3.40917968750000 -0.02253927805449

13.00000000000000 3.40917968750000 3.41015625000000 3.40966796875000 -0.01090018640774

14.00000000000000 3.40966796875000 3.41015625000000 3.40991210937500 -0.00507883521433

15.00000000000000 3.40991210937500 3.41015625000000 3.41003417968750 -0.00216770824306

16.00000000000000 3.41003417968750 3.41015625000000 3.41009521484375 -0.00071203190978

17.00000000000000 3.41009521484375 3.41015625000000 3.41012573242188 0.00001583446928

18.00000000000000 3.41009521484375 3.41012573242188 3.41011047363281 -0.00034810107130

19.00000000000000 3.41011047363281 3.41012573242188 3.41011810302734 -0.00016613388878

20.00000000000000 3.41011810302734 3.41012573242188 3.41012191772461 -0.00007514985669

21.00000000000000 3.41012191772461 3.41012573242188 3.41012382507324 -0.00002965773044

22.00000000000000 3.41012382507324 3.41012573242188 3.41012477874756 -0.00000691163977

由结果可知：x3=3.41012477874756 .

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