Equator

Index

0. Introduction
1. Phase 1: System of Linear Equations
   1. Gauss Elimination
   2. Gauss Jordan
   3. LU Decomposition
      1. Doolittle's Method
      2. Crout's Method
      3. Cholesky's Method
   4. Gauss Seidel
   5. Jacobi-Iteration
2. Phase 2: Root Finding
   1. Bisection
   2. False-Position
   3. Fixed Point
   4. Original Newton-Raphson
   5. Modified Newton-Raphson
      1. Modification 1
      2. Modification 2
   6. Secant Method

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Introduction

The project "Equator" stands as a robust mathematical toolkit, built upon the foundation of Python and PyQt, enriched by the inclusion of powerful libraries such as NumPy, SymPy, and Matplotlib. Its overarching goal is to deliver efficient numerical solutions to a diverse array of mathematical challenges, with a specific emphasis on problems related to linear algebra and root finding. The project is divided into two phases, each of which is further subdivided into several modules. The first phase is dedicated to the solution of systems of linear equations, while the second phase is dedicated to the solution of root finding problems. The project is designed to be user-friendly, with a simple and intuitive graphical user interface (GUI) that allows the user to easily interact with the program and obtain the desired results.

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Phase 1: System of Linear Equations

1. Gauss Elimination

$$ \begin{align*} &\text{Forward Elimination:} \\ &\begin{bmatrix} \begin{array}{ccc|c} a_{11} & a_{12} & a_{13} & b_1 \\ a_{21} & a_{22} & a_{23} & b_2 \\ a_{31} & a_{32} & a_{33} & b_3 \\ \end{array} \end{bmatrix} \sim \begin{bmatrix} \begin{array}{ccc|c} a_{11} & a_{12} & a_{13} & b_1 \\ 0 & a_{22}' & a_{23}' & b_2' \\ 0 & 0 & a_{33}'' & b_3'' \\ \end{array} \end{bmatrix}\\ &\text{Backward Substitution:} \\ &\begin{aligned} x_3 &= \frac{b_3''}{a_{33}''} \\ x_2 &= \frac{b_2' - a_{23}'x_3}{a_{22}'} \\ x_1 &= \frac{b_1 - a_{12}x_2 - a_{13}x_3}{a_{11}} \\ \end{aligned} \end{align*} $$

Sample Run

$$ \begin{aligned} 2x_1 + 3x_2 - 2x_3 &= 1 \\ 4x_1 + 4x_2 - 3x_3 &= 5 \\ 2x_1 - x_2 + 2x_3 &= 3 \\ \end{aligned} $$

Steps

Step 1: Step 2: Step 3: Step 4: Step 5: Step 6: Step 7: Step 8: Step 9: Step 10:

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2. Gauss Jordan

$$ \begin{align*} &\text{Forward and Backward Elimination:} \\ &\begin{bmatrix} \begin{array}{ccc|c} a_{11} & a_{12} & a_{13} & b_1 \\ a_{21} & a_{22} & a_{23} & b_2 \\ a_{31} & a_{32} & a_{33} & b_3 \\ \end{array} \end{bmatrix} \sim \begin{bmatrix} \begin{array}{ccc|c} a_{11} & 0 & 0 & b_1'' \\ 0 & a_{22}' & 0 & b_2'' \\ 0 & 0 & a_{33}'' & b_3'' \\ \end{array} \end{bmatrix} \ \\ &\text{Normalization:} \\ &\begin{bmatrix} \begin{array}{ccc|c} 1 & 0 & 0 & b_1''/a_{11} \\ 0 & 1 & 0 & b_2''/a_{22}' \\ 0 & 0 & 1 & b_3''/a_{33}'' \\ \end{array} \end{bmatrix} \begin{aligned} x_1 = \frac{b_1''}{a_{11}},\quad x_2 = \frac{b_2''}{a_{22}'} ,\quad x_3 = \frac{b_3''}{a_{33}''} \\ \end{aligned} \end{align*} $$

Sample Run

$$ \begin{aligned} 2x_1 + 3x_2 - 2x_3 &= 1 \\ 4x_1 + 4x_2 - 3x_3 &= 5 \\ 2x_1 - x_2 + 2x_3 &= 3 \\ \end{aligned} $$

Steps

Step 1: Step 2: Step 3: Step 4: Step 5: Step 6: Step 7: Step 8: Step 9:

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3. LU Decomposition

$$ A = LU \text{ where } L \text{ is a lower triangular matrix and } U \text{ is an upper triangular matrix}\\ $$

$$ LUx = b \implies Ux = y \text{ and } Ly = b $$

a. Doolittle's Method

$$ \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix} =\begin{bmatrix} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1 \\ \end{bmatrix} \begin{bmatrix} u_{11} & u_{12} & u_{13} \\ 0 & a_{22} & u_{23} \\ 0 & 0 & u_{33} \\ \end{bmatrix} $$

Sample Run

$$ \begin{aligned} 3x_1 + 2x_2 - x_3 &= 1 \\ 2x_1 + 3x_2 + 2x_3 &= 2 \\ 5x_1 - x_2 + 4x_3 &= 3 \\ \end{aligned} $$

Steps

Step 1: Step 2: Step 3: Step 4:

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b. Crout's Method

$$ \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix} =\begin{bmatrix} l_{11} & 0 & 0 \\ l_{21} & l_{22} & 0 \\ l_{31} & l_{32} & l_{33} \\ \end{bmatrix} \begin{bmatrix} 1 & u_{12} & u_{13} \\ 0 & 1 & u_{23} \\ 0 & 0 & 1 \\ \end{bmatrix} $$

Sample Run

$$ \begin{aligned} 3x_1 + 2x_2 - x_3 &= 1 \\ 2x_1 + 3x_2 + 2x_3 &= 2 \\ 5x_1 - x_2 + 4x_3 &= 3 \\ \end{aligned} $$

Steps

Step 1: Step 2: Step 3: Step 4:

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c. Cholesky's Method

$$ A = A^T \implies A = LL^T \\ $$

$$ \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix} =\begin{bmatrix} l_{11} & 0 & 0 \\ l_{21} & l_{22} & 0 \\ l_{31} & l_{32} & l_{33} \\ \end{bmatrix} \begin{bmatrix} l_{11} & l_{21} & l_{31} \\ 0 & l_{22} & l_{32} \\ 0 & 0 & l_{33} \\ \end{bmatrix} $$

Sample Run

$$ \begin{aligned} 5x_1 + 2x_2 + x_3 &= 1 \\ 2x_1 + 6x_2 + 2x_3 &= 2 \\ 3x_1 + 2x_2 + 7x_3 &= 3 \\ \end{aligned} $$

Steps

Step 1: Step 2: Step 3: Step 4: Step 5: Step 6: Step 7:

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4. Gauss Seidel

$$ \begin{aligned} x_1^{(k+1)} &= \frac{b_1 - a_{12}x_2^{(k)} - a_{13}x_3^{(k)}}{a_{11}} \\ x_2^{(k+1)} &= \frac{b_2 - a_{21}x_1^{(k+1)} - a_{23}x_3^{(k)}}{a_{22}} \\ x_3^{(k+1)} &= \frac{b_3 - a_{31}x_1^{(k+1)} - a_{32}x_2^{(k+1)}}{a_{33}} \\ \end{aligned} $$

Sample Run

$$ \begin{aligned} &4x_1 + 2x_2 + x_3 &= 11 \\ &-1x_1 + 2x_2 &= 3 \\ &2x_1 + x_2 + 4x_3 &= 16 \\ &x_0 = (1, 1, 1) \\ \end{aligned} $$

Steps

Step 1: Step 2: Step 3: Step 4: Step 5: Step 6:

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5. Jacobi-Iteration

$$ \begin{aligned} x_1^{(k+1)} &= \frac{b_1 - a_{12}x_2^{(k)} - a_{13}x_3^{(k)}}{a_{11}} \\ x_2^{(k+1)} &= \frac{b_2 - a_{21}x_1^{(k)} - a_{23}x_3^{(k)}}{a_{22}} \\ x_3^{(k+1)} &= \frac{b_3 - a_{31}x_1^{(k)} - a_{32}x_2^{(k)}}{a_{33}} \\ \end{aligned} $$

Sample Run

$$ \begin{aligned} &4x_1 + 2x_2 + x_3 &= 11 \\ &-1x_1 + 2x_2 &= 3 \\ &2x_1 + x_2 + 4x_3 &= 16 \\ &x_0 = (1, 1, 1) \\ \end{aligned} $$

Steps

Step 1: Step 2: Step 3: Step 4: Step 5: Step 6: Step 7: Step 8: Step 9: Step 10: Step 11:

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Phase 2: Root Finding

1. Bisection

$$ x_{r} = \frac{x_{l} + x_{u}}{2} \begin{cases} x_{u} = x_{r}, & \text{if } f(x_{l}) \cdot f(x_{r}) < 0 \\ x_{l} = x_{r}, & \text{if } f(x_{l}) \cdot f(x_{r}) > 0 \\ \text{root} = x_{\text{r}}, & \text{if } f(x_{l}) \cdot f(x_{r}) = 0 \end{cases} $$

Sample Run

$$ f(x) = -12 - 21x + 18x^2 - 2.75x^3 $$

Steps

Step 1: Step 2: Step 3: Step 4: Step 5: Step 6: Step 7: Step 8: Step 9:

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2. False-Position

$$ x_{r} = \frac{x_{l} \cdot f(x_{u}) - x_{u} \cdot f(x_{l})}{f(x_{u}) - f(x_{l})} \begin{cases} x_{u} = x_{r}, & \text{if } f(x_{l}) \cdot f(x_{r}) < 0 \\ x_{l} = x_{r}, & \text{if } f(x_{l}) \cdot f(x_{r}) > 0 \\ \text{root} = x_{\text{r}}, & \text{if } f(x_{l}) \cdot f(x_{r}) = 0 \end{cases} $$

Sample Run

$$ f(x) = -12 - 21x + 18x^2 - 2.75x^3 $$

Steps

Step 1: Step 2: Step 3: Step 4: Step 5: Step 6: Step 7:

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3. Fixed Point

$$ x_{i+1} = g(x_{i}) $$

Sample Run

$$ f(x) = \sin{\sqrt{x}} - x, \quad g(x) = \sin{\sqrt{x}} $$

Steps

Step 1: Step 2: Step 3: Step 4: Step 5: Step 6: Step 7: Step 8: Step 9:

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4. Original Newton-Raphson

$$ x_{i+1} = x_i - \frac{f(x_i)}{f'(x_i)} $$

Sample Run

$$ f(x) = -0.9x^2 + 1.7x + 2.5 $$

Steps

Step 1: Step 2: Step 3: Step 4: Step 5:

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5. Modified Newton-Raphson

Modification 1:

$$ x_{i+1} = x_i - m \cdot \frac{f(x_i)}{f'(x_i)} $$

Sample Run

$$ f(x) = x^3 - 5x^2 + 7x - 3 $$

Steps

Step 1: Step 2: Step 3: Step 4:

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Modification 2:

$$ x_{i+1} = x_i - \frac{f(x_i) \cdot f'(x_i)}{(f'(x_i))^2 - f(x_i) \cdot f''(x_i)} $$

Sample Run

$$ f(x) = x^3 - 5x^2 + 7x - 3 $$

Steps

Step 1: Step 2: Step 3: Step 4:

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6. Secant Method

$$ x_{i+1} = x_i - \frac{f(x_i) \cdot (x_i - x_{i-1})}{f(x_i) - f(x_{i-1})} $$

Sample Run

$$ f(x) = x^3 - 6x^2 + 11x - 6.1 $$

Steps

Step 1: Step 2: Step 3: