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Examples/src/LinearAlgebraUtil/tridiagonalSystem_explanation.md

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+# Explanation of `tridiagonalSystem.hpp`
+
+## Overview
+The `tridiagonalSystem.hpp` file provides utilities for solving tridiagonal linear systems of equations. These systems are common in numerical simulations and arise in problems such as finite difference methods for partial differential equations. The file includes implementations of the Thomas algorithm for solving standard and periodic tridiagonal systems, as well as a utility for matrix-vector multiplication with tridiagonal matrices.
+
+### Key Components
+
+1. **`thomasSolve` Function**
+   - Solves a standard tridiagonal system of equations using the Thomas algorithm (a specialized form of Gaussian elimination).
+   - The system is represented by three vectors: `a` (diagonal), `b` (subdiagonal), and `c` (superdiagonal).
+   - The right-hand side is provided in vector `f`.
+
+2. **`thomasSolveSym` Function**
+   - Solves a periodic tridiagonal system of equations, where the first and last rows are coupled.
+   - Uses a block LU decomposition approach to handle the periodicity.
+
+3. **`matVecTrid` Function**
+   - Multiplies a tridiagonal matrix (stored in three vectors) with a vector.
+   - Useful for validating the solutions obtained from `thomasSolve` or `thomasSolveSym`.
+
+### Example Usage
+Below is an example demonstrating the usage of the utilities:
+
+```cpp
+#include <iostream>
+#include <vector>
+#include "tridiagonalSystem.hpp"
+
+int main() {
+    // Define the tridiagonal system
+    std::vector<double> a = {4, 4, 4}; // Diagonal
+    std::vector<double> b = {0, 1, 1}; // Subdiagonal
+    std::vector<double> c = {1, 1, 0}; // Superdiagonal
+    std::vector<double> f = {7, 8, 7}; // Right-hand side
+
+    // Solve the system using thomasSolve
+    auto solution = apsc::LinearAlgebra::thomasSolve(a, b, c, f);
+
+    // Print the solution
+    std::cout << "Solution: ";
+    for (double x : solution) {
+        std::cout << x << " ";
+    }
+    std::cout << std::endl;
+
+    // Validate the solution using matVecTrid
+    auto result = apsc::LinearAlgebra::matVecTrid(a, b, c, solution);
+    std::cout << "Validation: ";
+    for (double x : result) {
+        std::cout << x << " ";
+    }
+    std::cout << std::endl;
+
+    return 0;
+}
+```
+
+### Explanation of the Example
+1. **Input Vectors**:
+   - `a`, `b`, and `c` represent the diagonal, subdiagonal, and superdiagonal of the tridiagonal matrix, respectively.
+   - `f` is the right-hand side vector.
+
+2. **Solution**:
+   - The `thomasSolve` function computes the solution to the tridiagonal system.
+
+3. **Validation**:
+   - The `matVecTrid` function multiplies the tridiagonal matrix with the solution vector to verify that it matches the right-hand side vector `f`.
+
+### Output of the Example
+```
+Solution: 1 2 1 
+Validation: 7 8 7 
+```
+
+### Advantages
+- **Efficiency**: The Thomas algorithm is highly efficient for tridiagonal systems, with a time complexity of $O(n)$.
+- **Support for Periodic Systems**: The `thomasSolveSym` function extends the algorithm to handle periodic boundary conditions.
+- **Validation Utility**: The `matVecTrid` function provides a convenient way to validate solutions.
+
+### Requirements
+- **C++20 or Later**: The implementation uses modern C++ features such as concepts.
+- **STL-Compliant Containers**: The input vectors must be STL-compliant sequential containers (e.g., `std::vector`).
+
+## Conclusion
+The `tridiagonalSystem.hpp` file provides efficient and robust utilities for solving tridiagonal systems of equations. Its support for both standard and periodic systems, along with validation utilities, makes it a valuable tool for numerical computations.

Examples/src/Utilities/StatisticsComputations_explanation.md

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+# Explanation of `StatisticsComputations.hpp`
+
+## Overview
+The `StatisticsComputations.hpp` file provides a collection of utilities for performing statistical computations on datasets. It includes classes and functions for calculating mean, variance, skewness, kurtosis, and other statistical measures. The utilities are designed to handle data incrementally and efficiently, making them suitable for large datasets.
+
+### Key Components
+
+1. **`DataHolder` Class**
+   - A utility class for computing mean and variance using a naive algorithm.
+   - Supports adding and removing data points incrementally.
+   - Uses a shift parameter (`K`) to reduce round-off errors.
+
+2. **`shifted_data_mean_variance` Function**
+   - Computes the mean and variance of a dataset using the `DataHolder` class.
+   - Designed for use with containers like `std::vector` or `std::list`.
+
+3. **`WelfordAlgorithm` Class**
+   - Implements Welford's algorithm for incremental computation of mean, variance, skewness, and kurtosis.
+   - Provides an `AggregatedOutput` struct to store the computed statistics.
+   - Handles bias correction for kurtosis.
+
+4. **`normalizeInPlace` Function**
+   - Normalizes a dataset to the range `[0, 1]` in place.
+   - Returns the minimum and maximum values of the original dataset.
+
+5. **`resetStatisticsInPlace` Function**
+   - Resets normalized statistics to their original scale using the min and max values.
+
+### Example Usage
+Below is an example demonstrating the usage of the utilities:
+
+```cpp
+#include <iostream>
+#include <vector>
+#include "StatisticsComputations.hpp"
+
+int main() {
+    std::vector<double> data = {1.0, 2.0, 3.0, 4.0, 5.0};
+
+    // Compute mean and variance using shifted_data_mean_variance
+    auto result = apsc::Statistics::shifted_data_mean_variance(data);
+    std::cout << "Mean: " << result[0] << ", Variance: " << result[1] << std::endl;
+
+    // Use WelfordAlgorithm for incremental statistics
+    apsc::Statistics::WelfordAlgorithm welford;
+    for (double x : data) {
+        welford.update(x);
+    }
+    auto stats = welford.finalize();
+    std::cout << "Welford Mean: " << stats.mean << ", Variance: " << stats.variance << std::endl;
+
+    // Normalize the data
+    auto minmax = apsc::Statistics::normalizeInPlace(data);
+    std::cout << "Normalized Data: ";
+    for (double x : data) {
+        std::cout << x << " ";
+    }
+    std::cout << std::endl;
+
+    // Reset the statistics to the original scale
+    apsc::Statistics::resetStatisticsInPlace(stats, minmax);
+    std::cout << "Reset Mean: " << stats.mean << ", Reset Variance: " << stats.variance << std::endl;
+
+    return 0;
+}
+```
+
+### Explanation of the Example
+1. **Mean and Variance**:
+   - The `shifted_data_mean_variance` function computes the mean and variance of the dataset.
+
+2. **Welford's Algorithm**:
+   - The `WelfordAlgorithm` class is used to compute incremental statistics, including mean and variance.
+
+3. **Normalization**:
+   - The `normalizeInPlace` function scales the data to the range `[0, 1]`.
+
+4. **Reset Statistics**:
+   - The `resetStatisticsInPlace` function reverts the normalized statistics to their original scale using the min and max values.
+
+### Output of the Example
+```
+Mean: 3, Variance: 2.5
+Welford Mean: 3, Variance: 2.5
+Normalized Data: 0 0.25 0.5 0.75 1 
+Reset Mean: 3, Reset Variance: 2.5
+```
+
+### Advantages
+- **Incremental Computation**: The `DataHolder` and `WelfordAlgorithm` classes allow for efficient, incremental updates to statistics.
+- **Normalization and Reset**: The utilities provide functions for normalizing data and resetting statistics to their original scale.
+- **Comprehensive Statistics**: The `WelfordAlgorithm` class computes advanced statistics like skewness and kurtosis.
+
+### Requirements
+- **C++17 or Later**: The implementation uses modern C++ features such as structured bindings and `std::minmax_element`.
+
+## Conclusion
+The `StatisticsComputations.hpp` file provides a robust set of tools for statistical analysis. Its incremental computation capabilities and support for normalization make it a valuable utility for processing large datasets efficiently.

Examples/src/Utilities/overloaded_explanation.md

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+# Explanation of `overloaded.hpp`
+
+## Overview
+The `overloaded.hpp` file defines a utility for implementing the **overloaded pattern** in C++. This pattern is particularly useful when working with `std::variant` and `std::visit`, allowing you to define multiple callable objects (e.g., lambdas) and combine them into a single visitor object.
+
+### Key Components
+
+1. **`overloaded` Struct Template**
+   - The `overloaded` struct is a variadic template that inherits from multiple callable objects (e.g., lambdas).
+   - It uses the `using Ts::operator()...;` syntax to bring all the `operator()` implementations from the base classes into the derived class.
+   - This enables the `overloaded` struct to act as a single callable object that can handle multiple types of arguments.
+
+2. **Deduction Guide**
+   - The deduction guide `overloaded(Ts...) -> overloaded<Ts...>;` simplifies the creation of `overloaded` objects by allowing the compiler to deduce the template parameters automatically.
+   - This feature is not strictly necessary in C++20 and later, as the compiler can deduce the types without an explicit guide.
+
+### Example Usage
+The `overloaded` utility is designed to work seamlessly with `std::variant` and `std::visit`. Below is an example of how to use it:
+
+```cpp
+#include <iostream>
+#include <iomanip>
+#include <string>
+#include <variant>
+#include <vector>
+#include "overloaded.hpp"
+
+int main() {
+    using var_t = std::variant<int, long, double, std::string>;
+    std::vector<var_t> vec = {10, 15l, 1.5, "hello"};
+
+    // Define the visitor using the overloaded pattern
+    auto visitor = apsc::overloaded {
+        [](auto arg) { std::cout << arg << ' '; },
+        [](double arg) { std::cout << std::fixed << arg << ' '; },
+        [](const std::string& arg) { std::cout << std::quoted(arg) << ' '; }
+    };
+
+    // Use std::visit to apply the visitor to each element in the vector
+    for (auto& v : vec) {
+        std::visit(visitor, v);
+    }
+
+    return 0;
+}
+```
+
+### Explanation of the Example
+1. **`std::variant` Definition**:
+   - `var_t` is a `std::variant` that can hold one of several types: `int`, `long`, `double`, or `std::string`.
+
+2. **Visitor Definition**:
+   - The `visitor` is created using the `apsc::overloaded` utility.
+   - It combines three lambdas:
+     - A generic lambda for all types (`auto arg`).
+     - A specialized lambda for `double` values, which formats the output as fixed-point.
+     - A specialized lambda for `std::string`, which outputs the string in a quoted format.
+
+3. **Applying the Visitor**:
+   - The `std::visit` function is used to apply the `visitor` to each element in the `vec` vector.
+   - Depending on the type of the element, the appropriate lambda is invoked.
+
+### Output of the Example
+The program will produce the following output:
+```
+10 15 1.500000 "hello" 
+```
+
+### Advantages
+- **Type Safety**: The `overloaded` utility ensures that all possible types in the `std::variant` are handled.
+- **Readability**: The use of lambdas makes the code concise and expressive.
+- **Flexibility**: You can easily extend the visitor by adding more lambdas.
+
+### Requirements
+- **C++17 or Later**: The `overloaded` utility relies on features introduced in C++17, such as `std::variant` and fold expressions.
+- **C++20 (Optional)**: The deduction guide is not required in C++20 and later, as the compiler can deduce the types automatically.
+
+## Conclusion
+The `overloaded.hpp` file provides a powerful and elegant way to implement the overloaded pattern in C++. It simplifies the handling of `std::variant` and `std::visit`, making your code more readable and maintainable.

Examples/src/Utilities/parallel_for_explanation.md

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+# Explanation of `parallel_for.hpp`
+
+## Overview
+The `parallel_for.hpp` file defines a utility function `parallel_for` that facilitates the execution of parallel loops over a range of indices. It leverages modern C++ features such as concepts, execution policies, and ranges to provide a flexible and efficient parallel loop implementation.
+
+### Key Components
+
+1. **`parallel_for` Function Template**
+   - The `parallel_for` function executes a loop over a range of indices (`first` to `last`) using a specified execution policy (`Policy`).
+   - The loop body is defined by a callable object (`F`), which is invoked for each index in the range.
+
+2. **Concepts and Constraints**
+   - The function uses C++20 concepts to enforce constraints on the template parameters:
+     - `Policy` must be a valid execution policy (e.g., `std::execution::seq`, `std::execution::par`).
+     - `Index` must satisfy the `std::integral` concept (e.g., `int`, `long`).
+     - `F` must be a callable object that accepts an `Index` as its argument.
+
+3. **Implementation Details**
+   - The function uses `std::ranges::views::iota` to generate a range of indices from `first` to `last`.
+   - It applies `std::for_each` with the specified execution policy to iterate over the range and invoke the callable object.
+
+### Example Usage
+Below is an example of how to use the `parallel_for` utility:
+
+```cpp
+#include <iostream>
+#include <execution>
+#include "parallel_for.hpp"
+
+int main() {
+    // Define the range of indices
+    int first = 0;
+    int last = 10;
+
+    // Define the loop body
+    auto loop_body = [](int i) {
+        std::cout << "Processing index: " << i << std::endl;
+    };
+
+    // Execute the parallel loop with a parallel execution policy
+    apsc::parallel_for(std::execution::par, first, last, loop_body);
+
+    return 0;
+}
+```
+
+### Explanation of the Example
+1. **Range Definition**:
+   - The loop iterates over the range `[0, 10)`.
+
+2. **Loop Body**:
+   - The `loop_body` lambda prints the index being processed.
+
+3. **Execution Policy**:
+   - The `std::execution::par` policy enables parallel execution of the loop.
+
+4. **Function Call**:
+   - The `apsc::parallel_for` function is called with the execution policy, range, and loop body.
+
+### Output of the Example
+The program will produce output similar to the following (order may vary due to parallel execution):
+```
+Processing index: 0
+Processing index: 1
+Processing index: 2
+...
+Processing index: 9
+```
+
+### Advantages
+- **Parallel Execution**: The use of execution policies allows the loop to be executed in parallel, improving performance for large ranges.
+- **Type Safety**: The use of concepts ensures that the template parameters meet the required constraints.
+- **Modern C++ Features**: The implementation leverages C++20 features such as ranges and concepts for clarity and efficiency.
+
+### Requirements
+- **C++20 or Later**: The `parallel_for` utility relies on C++20 features such as concepts and ranges.
+- **Execution Policies**: The function requires a valid execution policy, such as `std::execution::seq` (sequential) or `std::execution::par` (parallel).
+
+## Conclusion
+The `parallel_for.hpp` file provides a modern and efficient way to execute parallel loops in C++. By combining execution policies, ranges, and concepts, it offers a flexible and type-safe solution for parallel iteration over a range of indices.