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问题描述
我不确定我做错了什么。修改后应该更快收敛。尝试更大的迭代时遇到了同样的问题。
这是我原来的：
I am unsure what I have done wrong. The modified should definitely converge faster. I had the same issues trying larger iterations.
Here is my original:
#include <stdio.h>
#include <math.h>
double f(double x)
{
//return (x*x-6);
//return (x*x*x-2*x*x-5);
//return (x*x-2*x*exp(-1*x)+exp(-2*x));
return (cos(x+sqrt(2))+x*((x/2)+sqrt(2)));
}
double fp(double x)
{
//return (2*x);
//return (3*x*x-4*x);
//return (2*x+2*x*exp(-1*x)-2*exp(-1*x)-2*exp(-2*x));
return (x-sin(x+sqrt(2))+sqrt(2));
}
int main()
{
double p0, TOL, p;
int i=1, N0;
printf("Initial approximation: ");
scanf("%lf", &p0);
printf("Desired Tolerance: ");
scanf("%lf", &TOL);
printf("Maximum number of iterations: ");
scanf("%d", &N0);
while(i<=N0)
{
p=p0-(f(p0)/fp(p0));
if((fabs(p-p0)) < TOL)
{
printf("Iteration: %d. Current value= %lf.", i, p);
}
i++;
p0=p;
}
if(i>N0)
printf("Method failed after %d iterations.", N0);
}
结果：
Results:
Initial approximation: -1.5
Desired Tolerance: .01
Maximum number of iterations: 100
Iteration: 2. Current value= -1.414242.Iteration: 3. Current value= -1.414251.
Iteration: 4. Current value= -1.414263.Iteration: 5. Current value= -1.414280.
Iteration: 6. Current value= -1.414302.Iteration: 7. Current value= -1.414331.
Iteration: 8. Current value= -1.414368.Iteration: 9. Current value= -1.414404.
Iteration: 10. Current value= -1.414444.Iteration: 11. Current value= -1.41444
5.Iteration: 12. Current value= -1.414462.Iteration: 13. Current value= -1.414
457.Iteration: 14. Current value= -1.414463.Iteration: 15. Current value= -1.4
14442.Iteration: 16. Current value= -1.414447.Iteration: 17. Current value= -1
.414442.Iteration: 18. Current value= -1.414457.Iteration: 19. Current value=
-1.414436.Iteration: 20. Current value= -1.414444.Iteration: 21. Current value
= -1.414461.Iteration: 22. Current value= -1.414450.Iteration: 23. Current val
ue= -1.414452.Iteration: 24. Current value= -1.414448.Iteration: 25. Current v
alue= -1.414459.Iteration: 26. Current value= -1.414460.Iteration: 27. Current
value= -1.414469.Iteration: 28. Current value= -1.414438.Iteration: 29. Curre
nt value= -1.414442.Iteration: 30. Current value= -1.414439.Iteration: 31. Cur
rent value= -1.414465.Iteration: 32. Current value= -1.414448.Iteration: 33. C
urrent value= -1.414448.Iteration: 34. Current value= -1.414434.Iteration: 35.
Current value= -1.414455.Iteration: 36. Current value= -1.414469.Iteration: 37
. Current value= -1.414435.Iteration: 38. Current value= -1.414459.Iteration:
39. Current value= -1.414442.Iteration: 40. Current value= -1.414466.Iteration
: 41. Current value= -1.414441.Iteration: 42. Current value= -1.414456.Iterati
on: 43. Current value= -1.414442.Iteration: 44. Current value= -1.414424.Itera
tion: 45. Current value= -1.414454.Iteration: 46. Current value= -1.414466.Ite
ration: 47. Current value= -1.414413.Iteration: 48. Current value= -1.414441.I
teration: 49. Current value= -1.414437.Iteration: 50. Current value= -1.414457
.Iteration: 51. Current value= -1.414462.Iteration: 52. Current value= -1.4144
62.Iteration: 53. Current value= -1.414457.Iteration: 54. Current value= -1.41
4446.Iteration: 55. Current value= -1.414454.Iteration: 56. Current value= -1.
414466.Iteration: 57. Current value= -1.414460.Iteration: 58. Current value= -
1.414468.Iteration: 59. Current value= -1.414460.Iteration: 60. Current value=
-1.414434.Iteration: 61. Current value= -1.414436.Iteration: 62. Current valu
e= -1.414438.Iteration: 63. Current value= -1.414432.Iteration: 64. Current va
lue= -1.414435.Iteration: 65. Current value= -1.414436.Iteration: 66. Current
value= -1.414449.Iteration: 67. Current value= -1.414468.Iteration: 68. Curren
t value= -1.414468.Iteration: 69. Current value= -1.414448.Iteration: 70. Curr
ent value= -1.414467.Iteration: 71. Current value= -1.414466.Iteration: 72. Cu
rrent value= -1.414456.Iteration: 73. Current value= -1.414470.Iteration: 74.
Current value= -1.414437.Iteration: 75. Current value= -1.414439.Iteration: 76.
Current value= -1.414456.Iteration: 77. Current value= -1.414434.Iteration: 7
8. Current value= -1.414427.Iteration: 79. Current value= -1.414426.Iteration:
80. Current value= -1.414460.Iteration: 81. Current value= -1.414465.Iteratio
n: 82. Current value= -1.414420.Iteration: 83. Current value= -1.414456.Iterat
ion: 84. Current value= -1.414456.Iteration: 85. Current value= -1.414469.Iter
ation: 86. Current value= -1.414468.Iteration: 87. Current value= -1.414447.It
eration: 88. Current value= -1.414422.Iteration: 89. Current value= -1.414448.
Iteration: 90. Current value= -1.414442.Iteration: 91. Current value= -1.41445
1.Iteration: 92. Current value= -1.414464.Iteration: 93. Current value= -1.414
466.Iteration: 94. Current value= -1.414417.Iteration: 95. Current value= -1.4
14444.Iteration: 96. Current value= -1.414429.Iteration: 97. Current value= -1
.414453.Iteration: 98. Current value= -1.414434.Iteration: 99. Current value=
-1.414438.Iteration: 100. Current value= -1.414436.Method failed after 100 iter
ations.
Process returned 35 (0x23) execution time : 8.484 s
Press any key to continue.
这是修改过的牛顿：
Here is the modified Newtons:
#include <stdio.h>
#include <math.h>
double f(double x)
{
//return (x*x-2*x*exp(-1*x)+exp(-2*x));
return (cos(x+sqrt(2))+x*((x/2)+sqrt(2)));
}
double fp(double x)
{
//return (2*x+2*x*exp(-1*x)-2*exp(-1*x)-2*exp(-2*x));
return (x-sin(x+sqrt(2))+sqrt(2));
}
double fpp(double x)
{
//return (2-2*x*exp(-1*x)+4*exp(-1*x)+4*exp(-2*x));
return (1-cos(x+sqrt(2)));
}
int main()
{
double p0, TOL, p;
int i=1, N0;
printf("Initial approximation: ");
scanf("%lf", &p0);
printf("Desired Tolerance: ");
scanf("%lf", &TOL);
printf("Maximum number of iterations: ");
scanf("%d", &N0);
while(i<=N0)
{
p=p0-((f(p0)*fp(p0))/(pow(fp(p0),2)-f(p0)*fpp(p0)));
if((fabs(p-p0)) < TOL)
{
printf("Iteration: %d. Current value= %lf.", i, p);
}
i++;
p0=p;
}
if(i>N0)
printf("Method failed after %d iterations.", N0);
}
Results:
Results:
Initial approximation: -1.5
Desired Tolerance: .01
Maximum number of iterations: 100
Iteration: 2. Current value= -1.414242.Iteration: 3. Current value= -1.414251.
Iteration: 4. Current value= -1.414263.Iteration: 5. Current value= -1.414280.
Iteration: 6. Current value= -1.414302.Iteration: 7. Current value= -1.414331.
Iteration: 8. Current value= -1.414368.Iteration: 9. Current value= -1.414404.
Iteration: 10. Current value= -1.414444.Iteration: 11. Current value= -1.41444
5.Iteration: 12. Current value= -1.414462.Iteration: 13. Current value= -1.414
457.Iteration: 14. Current value= -1.414463.Iteration: 15. Current value= -1.4
14442.Iteration: 16. Current value= -1.414447.Iteration: 17. Current value= -1
.414442.Iteration: 18. Current value= -1.414457.Iteration: 19. Current value=
-1.414436.Iteration: 20. Current value= -1.414444.Iteration: 21. Current value
= -1.414461.Iteration: 22. Current value= -1.414450.Iteration: 23. Current val
ue= -1.414452.Iteration: 24. Current value= -1.414448.Iteration: 25. Current v
alue= -1.414459.Iteration: 26. Current value= -1.414460.Iteration: 27. Current
value= -1.414469.Iteration: 28. Current value= -1.414438.Iteration: 29. Curre
nt value= -1.414442.Iteration: 30. Current value= -1.414439.Iteration: 31. Cur
rent value= -1.414465.Iteration: 32. Current value= -1.414448.Iteration: 33. C
urrent value= -1.414448.Iteration: 34. Current value= -1.414434.Iteration: 35.
Current value= -1.414455.Iteration: 36. Current value= -1.414469.Iteration: 37
. Current value= -1.414435.Iteration: 38. Current value= -1.414459.Iteration:
39. Current value= -1.414442.Iteration: 40. Current value= -1.414466.Iteration
: 41. Current value= -1.414441.Iteration: 42. Current value= -1.414456.Iterati
on: 43. Current value= -1.414442.Iteration: 44. Current value= -1.414424.Itera
tion: 45. Current value= -1.414454.Iteration: 46. Current value= -1.414466.Ite
ration: 47. Current value= -1.414413.Iteration: 48. Current value= -1.414441.I
teration: 49. Current value= -1.414437.Iteration: 50. Current value= -1.414457
.Iteration: 51. Current value= -1.414462.Iteration: 52. Current value= -1.4144
62.Iteration: 53. Current value= -1.414457.Iteration: 54. Current value= -1.41
4446.Iteration: 55. Current value= -1.414454.Iteration: 56. Current value= -1.
414466.Iteration: 57. Current value= -1.414460.Iteration: 58. Current value= -
1.414468.Iteration: 59. Current value= -1.414460.Iteration: 60. Current value=
-1.414434.Iteration: 61. Current value= -1.414436.Iteration: 62. Current valu
e= -1.414438.Iteration: 63. Current value= -1.414432.Iteration: 64. Current va
lue= -1.414435.Iteration: 65. Current value= -1.414436.Iteration: 66. Current
value= -1.414449.Iteration: 67. Current value= -1.414468.Iteration: 68. Curren
t value= -1.414468.Iteration: 69. Current value= -1.414448.Iteration: 70. Curr
ent value= -1.414467.Iteration: 71. Current value= -1.414466.Iteration: 72. Cu
rrent value= -1.414456.Iteration: 73. Current value= -1.414470.Iteration: 74.
Current value= -1.414437.Iteration: 75. Current value= -1.414439.Iteration: 76.
Current value= -1.414456.Iteration: 77. Current value= -1.414434.Iteration: 7
8. Current value= -1.414427.Iteration: 79. Current value= -1.414426.Iteration:
80. Current value= -1.414460.Iteration: 81. Current value= -1.414465.Iteratio
n: 82. Current value= -1.414420.Iteration: 83. Current value= -1.414456.Iterat
ion: 84. Current value= -1.414456.Iteration: 85. Current value= -1.414469.Iter
ation: 86. Current value= -1.414468.Iteration: 87. Current value= -1.414447.It
eration: 88. Current value= -1.414422.Iteration: 89. Current value= -1.414448.
Iteration: 90. Current value= -1.414442.Iteration: 91. Current value= -1.41445
1.Iteration: 92. Current value= -1.414464.Iteration: 93. Current value= -1.414
466.Iteration: 94. Current value= -1.414417.Iteration: 95. Current value= -1.4
14444.Iteration: 96. Current value= -1.414429.Iteration: 97. Current value= -1
.414453.Iteration: 98. Current value= -1.414434.Iteration: 99. Current value=
-1.414438.Iteration: 100. Current value= -1.414436.Method failed after 100 iter
ations.
Process returned 35 (0x23) execution time : 8.484 s
Press any key to continue.推荐答案
printf("Iteration: %d. Current value= %lf.", i, p);
if((fabs(p-p0)) < TOL)
{
break;
}
secondly, I think you forgot to compile one of your codes before pasting output into the question. both outputs are the same, which is not expected.
third, tolerance value is too big. you may get a better output with 0.0001 since the root is about -1.4144.
last, could you provide a website for the proof of the second approximation? (this part is for me. please? :) )
secondly, I think you forgot to compile one of your codes before pasting output into the question. both outputs are the same, which is not expected.
third, tolerance value is too big. you may get a better output with 0.0001 since the root is about -1.4144.
last, could you provide a website for the proof of the second approximation? (this part is for me. please? :) )
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