Procedures

backward algorithm, solves the system $$U * x = y$$ where U is an upper triangular matrix and y is a vector

Cholesky decomposition of a matrix A $$A = LL^T$$ This subroutine performs Cholesky decomposition of a given symmetric positive definite matrix A, where L is a lower triangular matrix.

function that calculates the cross product between two real 3-dimensional vectors $\vec{a}$ and $\vec{b}$ $$\vec{a} \times \vec{b}$$ ¹

function that extracts the diagonal of a matrix $$D = \begin{pmatrix} A(1,1) & 0 & \cdots & 0 \\ 0 & A(2,2) & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & A(n,n) \end{pmatrix}$$ where $D$ is a vector containing the diagonal elements of the matrix $A$.

function which checks if A is diagonally dominant $$\forall i, |A(i,i)| \geq \sum_{j \neq i} |A(i,j)|$$

function that calculates the dot product of two real 3-dimensional vectors $\vec{a}$ and $\vec{b}$ $$\vec{a} \cdot \vec{b}$$

Computes the eigenvalues and eigenvectors of a matrix A $$A * \vec{v} = \lambda * \vec{v}$$ with A a square matrix, λ the eigenvalue, and v the eigenvector. This subroutine allows you to choose the method for computing eigenvalues and eigenvectors:

Perform a 1D Fourier Transform on a signal

Perform a 2D Fast Fourier Transform on a signal

Perform a 3D Fast Fourier Transform on a signal

forward algorithm, solves the system $$L * y = b$$ where L is a lower triangular matrix and b is a vector

function that returns the identity matrix for a given size N $$I_N = \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{pmatrix}$$

Perform a 1D inverse Fast Fourier Transform on a signal

Perform a 2D inverse Fast Fourier Transform on a signal

Perform a 3D inverse Fast Fourier Transform on a signal

Incomplete LU decomposition of a matrix A $$A \approx LU$$ This subroutine performs incomplete LU decomposition of a given matrix A, where L is a lower triangular matrix and U is an upper triangular matrix.

Incomplete Cholesky decomposition of a matrix A $$A \approx LL^T$$ This subroutine performs incomplete Cholesky decomposition of a given matrix A, where L is a lower triangular matrix and U is an upper triangular matrix.

Alternative Cholesky decomposition of a matrix A $$A = LDL^T$$ This subroutine performs alternative Cholesky decomposition of a given symmetric positive definite matrix A, where L is a lower triangular matrix and D is a diagonal matrix.

LDU decomposition of a matrix A $$A = LDU$$ This subroutine performs LDU decomposition of a given matrix A, where L is a lower triangular matrix, D is a diagonal matrix, and U is an upper triangular matrix.

LU decomposition of a matrix A $$A = LU$$ This subroutine performs LU decomposition of a given matrix A, where L is a lower triangular matrix and U is an upper triangular matrix.

Make N-dimensional meshgrid from two vectors x_vector and y_vector

function that calculates the Euclidean norm (L2 norm or modulus) of a vector $\vec{a}$, where $\vec{a} \in \mathbb{C}^n$ $$||\vec{a}||_2 = \sqrt{\sum_{i=1}^{n} |a_i|^2} \quad \text{ with } \quad \sum_{i=1}^{n} |a_i|^2 = \vec{a} \cdot \overline{\vec{a}}$$ where $n$ is the dimension of the complex vector $\vec{a}$.

function that calculates the Euclidean norm (L2 norm) of a vector $\vec{a}$, where $\vec{a} \in \mathbb{R}^n$ $$||\vec{a}||_2 = \sqrt{\sum_{i=1}^{n} a_i^2} \quad \text{ with } \quad \sum_{i=1}^{n} a_i^2 = \vec{a} \cdot \vec{a}$$ where $n$ is the dimension of the real vector $\vec{a}$.

function that normalises a real vector a to make it a unit vector, where $\vec{a} \in \mathbb{R}^n$ $$\hat{a} = \frac{\vec{a}}{||\vec{a}||_2}$$

function that normalises a complex vector a to make it a unit vector, where $\vec{a} \in \mathbb{C}^n$ $$\hat{a} = \frac{\vec{a}}{||\vec{a}||_2}$$

QR decomposition of a matrix A using various methods $$A = QR$$ This subroutine performs QR decomposition of a given matrix A using the specified method (Householder, Givens, Classical Gram-Schmidt, or Modified Gram-Schmidt). The output matrices Q is an orthogonal matrix and R is an upper triangular matrix.

QR decomposition using Givens rotations

QR decomposition using Classical Gram-Schmidt method

QR decomposition using Modified Gram-Schmidt method

QR decomposition using Householder method

Function to create a rotation matrix

function that calculates the trace of a square matrix $A$ $$\text{Tr}(A) = \sum_{i=1}^{n} A(i,i)$$