## Friday, November 10, 2023

### 🔍📊 Summary of the Code with Emojis for FindRoot.C in SWMM5

🔍📊 Summary of the Code with Emojis

This code snippet focuses on finding the roots of a function, i.e., solving for `func(x) = 0`, using two distinct methods: Newton-Raphson method and Ridder's Method. Authored by L. Rossman on 11/19/13, it's a part of a larger numerical analysis routine.

### 1. Newton-Raphson Method 📐🔢

• Purpose: Used to find the root of a function bracketed between `x1` and `x2`.
• Process: Combines Newton-Raphson and bisection methods.
• Root Refinement: Refines the root `rts` within an accuracy of `+/-xacc`.
• Function Evaluations: Returns the number of evaluations used or 0 if maximum iterations are exceeded.
• Constraints: Requires that `func(x1)` and `func(x2)` have opposite signs.
• Adjustments: May need to recalibrate subcatchment's properties like Percent Impervious and Width after LID placement.
• Special Conditions: Handles cases where Green Roofs and Roof Disconnection only treat direct precipitation.

### 2. Ridder's Method 🧮📉

• Functionality: Finds the root of `func` between `x1` and `x2` within an accuracy of `xacc`.
• Return Values: Returns the root if found, or a large negative number if not.
• Methodology: Uses an algorithmic approach to refine the root estimate.
• Conditions: Effective when `func(x1)` and `func(x2)` have opposite signs.
• Iterations: Executes up to a maximum of `MAXIT` iterations.
• Result: If the method converges, it returns the estimated root value.

### Common Aspects of Both Methods 🔍🔄

• Max Iterations: Both methods limit the number of iterations to `MAXIT` (60).
• Accuracy: Aim to refine the root estimate to within a specified accuracy (`xacc`).
• Error Checking: Includes checks for function values and bisection requirements.

### Implementation Details 🖥️💡

• Header File: Utilizes a custom header file `findroot.h`.
• Mathematical Functions: Employs standard math functions from `math.h`.
• Constants: Defines a `SIGN` function and sets `MAXIT` as 60.

### Application Scenario 🌍📚

This code is typically used in scenarios like hydrological modeling or environmental simulations where root-finding algorithms are crucial for analyzing and predicting natural processes.

🔎 In summary, the code effectively implements two root-finding algorithms with careful considerations for accuracy, iteration limits, and method-specific conditions, making it a valuable tool for numerical computations in various scientific and engineering applications.