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Zhe Yuan
2025-11-23 16:16:53 -05:00
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============================================================
Problem (a): cos(x) - x, x0=0.5
============================================================
Method: Newton
Iter | Value (x_k) | Error/Resid
------------------------------------------------------------
0 | 0.5 | 2.390851e-01
1 | 0.7552224171 | 1.613728e-02
2 | 0.7391416661 | 5.653293e-05
3 | 0.7390851339 | 7.056461e-10
4 | 0.7390851332 | 0.000000e+00
5 | 0.7390851332 | 0.000000e+00
Method: Chord
Iter | Value (x_k) | Error/Resid
------------------------------------------------------------
0 | 0.5 | 2.390851e-01
1 | 0.7552224171 | 1.613728e-02
2 | 0.7369022576 | 2.182876e-03
3 | 0.7393704622 | 2.853290e-04
4 | 0.7390476612 | 3.747205e-05
...
13 | 0.7390851332 | 4.333200e-13
14 | 0.7390851332 | 5.695444e-14
Method: Secant
Iter | Value (x_k) | Error/Resid
------------------------------------------------------------
0 | 0.5 | 2.390851e-01
1 | 0.7556018135 | 1.651668e-02
2 | 0.738106944 | 9.781892e-04
3 | 0.7390815948 | 3.538433e-06
4 | 0.739085134 | 7.646579e-10
5 | 0.7390851332 | 5.551115e-16
6 | 0.7390851332 | 0.000000e+00
============================================================
Problem (b): sin(x), x0=3.0
============================================================
Method: Newton
Iter | Value (x_k) | Error/Resid
------------------------------------------------------------
0 | 3 | 1.415927e-01
1 | 3.142546543 | 9.538895e-04
2 | 3.141592653 | 2.893161e-10
3 | 3.141592654 | 0.000000e+00
4 | 3.141592654 | 0.000000e+00
Method: Chord
Iter | Value (x_k) | Error/Resid
------------------------------------------------------------
0 | 3 | 1.415927e-01
1 | 3.142546543 | 9.538895e-04
2 | 3.141583011 | 9.642404e-06
3 | 3.141592751 | 9.747184e-08
4 | 3.141592653 | 9.853101e-10
5 | 3.141592654 | 9.960477e-12
6 | 3.141592654 | 1.003642e-13
7 | 3.141592654 | 1.332268e-15
Method: Secant
Iter | Value (x_k) | Error/Resid
------------------------------------------------------------
0 | 3 | 1.415927e-01
1 | 3.142873442 | 1.280788e-03
2 | 3.141588403 | 4.250744e-06
3 | 3.141592654 | 1.158629e-12
4 | 3.141592654 | 0.000000e+00
5 | 3.141592654 | 0.000000e+00
============================================================
Problem (c): x^2 + 1, x0=0.5
============================================================
Method: Newton
Iter | Value (x_k) | Error/Resid
------------------------------------------------------------
0 | 0.5 | 1.250000e+00
1 | -0.75 | 1.562500e+00
2 | 0.2916666667 | 1.085069e+00
3 | -1.568452381 | 3.460043e+00
4 | -0.4654406117 | 1.216635e+00
...
18 | -1.820817243 | 4.315375e+00
19 | -0.6358066524 | 1.404250e+00
Method: Chord
Iter | Value (x_k) | Error/Resid
------------------------------------------------------------
0 | 0.5 | 1.250000e+00
1 | -0.75 | 1.562500e+00
2 | -2.3125 | 6.347656e+00
3 | -8.66015625 | 7.599831e+01
4 | -84.65846252 | 7.168055e+03
...
8 | -7.660265913e+30 | 5.867967e+61
9 | -5.867967385e+61 | 3.443304e+123
Method: Secant
Iter | Value (x_k) | Error/Resid
------------------------------------------------------------
0 | 0.5 | 1.250000e+00
1 | -0.756281407 | 1.571962e+00
2 | 5.37745098 | 2.991698e+01
3 | -1.096446714 | 2.202195e+00
4 | -1.610857647 | 3.594862e+00
...
18 | 0.3478146096 | 1.120975e+00
19 | -2.615940634 | 7.843145e+00