My implementation of various computational methods for numerical analysis from our CSC 225 course.
- Bisection method
- Newton-Raphson method
- Secant method
- Forward Difference method
- Backward Difference method
- Center Difference method
- Trapezoidal Rule
- Euler's method
- Improved Euler's method (Heun's)
- Linear Spline Interpolation (Lerp)
- Quadratic Spline Interpolation
- Cubic Spline Interpolation (Cerp)
- Discrete Fourier Transforms (DFT)
- Fast Fourier Transform (FFT) - Cooley-Tukey
- LU Decomposition (Dolittle's)
- Gaussian Elimination
- Jacobi Method
EULER'S METHOD APPROXIMATION OF SOLUTIONS TO ODEs
Problem: dy/dx = y
Initial condition: y(0) = 1
--------------------------------------------------------
n | xn | yn | dy/dx | h | yn+1
--------------------------------------------------------
0 | 0.0000 | 1.0000 | 1.0000 | 0.1000 | 1.1000
1 | 0.1000 | 1.1000 | 1.1000 | 0.1000 | 1.2100
2 | 0.2000 | 1.2100 | 1.2100 | 0.1000 | 1.3310
3 | 0.3000 | 1.3310 | 1.3310 | 0.1000 | 1.4641
4 | 0.4000 | 1.4641 | 1.4641 | 0.1000 | 1.6105
5 | 0.5000 | 1.6105 | 1.6105 | 0.1000 | 1.7716
6 | 0.6000 | 1.7716 | 1.7716 | 0.1000 | 1.9487
7 | 0.7000 | 1.9487 | 1.9487 | 0.1000 | 2.1436
8 | 0.8000 | 2.1436 | 2.1436 | 0.1000 | 2.3579
9 | 0.9000 | 2.3579 | 2.3579 | 0.1000 | 2.5937
10 | 1.0000 | 2.5937 | 2.5937 | 0.1000 | 2.8531
11 | 1.1000 | 2.8531 | 2.8531 | 0.1000 | 3.1384
12 | 1.2000 | 3.1384 | 3.1384 | 0.1000 | 3.4523
13 | 1.3000 | 3.4523 | 3.4523 | 0.1000 | 3.7975
14 | 1.4000 | 3.7975 | 3.7975 | 0.1000 | 4.1772
15 | 1.5000 | 4.1772 | 4.1772 | 0.1000 | 4.5950
16 | 1.6000 | 4.5950 | 4.5950 | 0.1000 | 5.0545
17 | 1.7000 | 5.0545 | 5.0545 | 0.1000 | 5.5599
18 | 1.8000 | 5.5599 | 5.5599 | 0.1000 | 6.1159
19 | 1.9000 | 6.1159 | 6.1159 | 0.1000 | 6.7275
Problem: dy/dx = x
Initial condition: y(0) = 0
--------------------------------------------------------
n | xn | yn | dy/dx | h | yn+1
--------------------------------------------------------
0 | 0.0000 | 0.0000 | 0.0000 | 0.1000 | 0.0000
1 | 0.1000 | 0.0000 | 0.1000 | 0.1000 | 0.0100
2 | 0.2000 | 0.0100 | 0.2000 | 0.1000 | 0.0300
3 | 0.3000 | 0.0300 | 0.3000 | 0.1000 | 0.0600
4 | 0.4000 | 0.0600 | 0.4000 | 0.1000 | 0.1000
5 | 0.5000 | 0.1000 | 0.5000 | 0.1000 | 0.1500
6 | 0.6000 | 0.1500 | 0.6000 | 0.1000 | 0.2100
7 | 0.7000 | 0.2100 | 0.7000 | 0.1000 | 0.2800
8 | 0.8000 | 0.2800 | 0.8000 | 0.1000 | 0.3600
9 | 0.9000 | 0.3600 | 0.9000 | 0.1000 | 0.4500
10 | 1.0000 | 0.4500 | 1.0000 | 0.1000 | 0.5500
11 | 1.1000 | 0.5500 | 1.1000 | 0.1000 | 0.6600
12 | 1.2000 | 0.6600 | 1.2000 | 0.1000 | 0.7800
13 | 1.3000 | 0.7800 | 1.3000 | 0.1000 | 0.9100
14 | 1.4000 | 0.9100 | 1.4000 | 0.1000 | 1.0500
15 | 1.5000 | 1.0500 | 1.5000 | 0.1000 | 1.2000
16 | 1.6000 | 1.2000 | 1.6000 | 0.1000 | 1.3600
17 | 1.7000 | 1.3600 | 1.7000 | 0.1000 | 1.5300
18 | 1.8000 | 1.5300 | 1.8000 | 0.1000 | 1.7100
19 | 1.9000 | 1.7100 | 1.9000 | 0.1000 | 1.9000
Problem: dy/dx = sin(x)
Initial condition: y(0) = 1
--------------------------------------------------------
n | xn | yn | dy/dx | h | yn+1
--------------------------------------------------------
0 | 0.0000 | 1.0000 | 0.0000 | 0.1000 | 1.0000
1 | 0.1000 | 1.0000 | 0.0998 | 0.1000 | 1.0100
2 | 0.2000 | 1.0100 | 0.1987 | 0.1000 | 1.0299
3 | 0.3000 | 1.0299 | 0.2955 | 0.1000 | 1.0594
4 | 0.4000 | 1.0594 | 0.3894 | 0.1000 | 1.0983
5 | 0.5000 | 1.0983 | 0.4794 | 0.1000 | 1.1463
6 | 0.6000 | 1.1463 | 0.5646 | 0.1000 | 1.2028
7 | 0.7000 | 1.2028 | 0.6442 | 0.1000 | 1.2672
8 | 0.8000 | 1.2672 | 0.7174 | 0.1000 | 1.3389
9 | 0.9000 | 1.3389 | 0.7833 | 0.1000 | 1.4172
10 | 1.0000 | 1.4172 | 0.8415 | 0.1000 | 1.5014
11 | 1.1000 | 1.5014 | 0.8912 | 0.1000 | 1.5905
12 | 1.2000 | 1.5905 | 0.9320 | 0.1000 | 1.6837
13 | 1.3000 | 1.6837 | 0.9636 | 0.1000 | 1.7801
14 | 1.4000 | 1.7801 | 0.9854 | 0.1000 | 1.8786
15 | 1.5000 | 1.8786 | 0.9975 | 0.1000 | 1.9784
16 | 1.6000 | 1.9784 | 0.9996 | 0.1000 | 2.0783
17 | 1.7000 | 2.0783 | 0.9917 | 0.1000 | 2.1775
18 | 1.8000 | 2.1775 | 0.9738 | 0.1000 | 2.2749
19 | 1.9000 | 2.2749 | 0.9463 | 0.1000 | 2.3695
LINEAR INTERPOLATION (LERP) VIA LINEAR SPLINES
----------------------------------------------
x: [0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, 2.8]
y: [10.0, 11.216, 11.728, 11.632, 11.024, 10.0, 8.656, 7.088, 5.392, 3.664, 2.0, 0.496, -0.752, -1.648, -2.096]
X: [0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, 2.8]
Y: [10.0, 11.216, 11.728, 11.632, 11.024, 10.0, 8.656, 7.088, 5.392, 3.664, 2.0, 0.496, -0.752, -1.648, -2.096]
DISCRETE FOURIER TRANSFORM (DFT) ON EVENLY SPACED SAMPLED SIGNALS
------------------------------------------------------------------
Sampling Rate: 20Hz per second
Sampled Time: [0.0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95]
Sampled Signals: [0.0, 5.57590997, 2.04087031, -8.37717508, -0.50202854, 10.0, -5.20431056, -0.768722952, -5.56758182, 10.278192, 1.71450552e-15, -10.278192, 5.56758182, 0.768722952, 5.20431056, -10.0, 0.50202854, 8.37717508, -2.04087031, -5.57590997]
Frequencies: [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0]
Magnitudes: [8.881784197001252e-16, 5.324005902807683e-09, 7.95147192533863e-09, 1.6912485634179598e-08, 2.8537386675967933e-08, 70.00000000399999, 2.1372325242942584e-08, 2.612710048506525e-09, 40.00000001048071, 30.000000004975725, 2.2942475566164232e-14, 30.00000000497574, 40.00000001048064, 2.612764227198249e-09, 2.1372252412524513e-08, 70.00000000400001, 2.8537468388468907e-08, 1.6912498068739307e-08, 7.951387104343746e-09, 5.3240201140351945e-09]
