NAFPack_matricielle Module

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

Arguments

Return Value real(kind=dp)

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

Arguments

Return Value real(kind=dp), dimension(3)

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 $\vec{a}$.

Arguments

Return Value real(kind=dp)

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 $\vec{a}$.

Arguments

Return Value real(kind=dp)

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

Arguments

Return Value real(kind=dp), dimension(size(a))

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

Arguments

Return Value complex(kind=dp), dimension(size(a))

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

Arguments

Return Value real(kind=dp)

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

Arguments

Return Value logical

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

Arguments

Return Value real(kind=dp), dimension(N, N)

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 $A$.

Arguments

Return Value real(kind=dp), dimension(size(A, 1))

Arguments

Return Value real(kind=dp), dimension(size(d, 1), size(d, 1))

Function to create a rotation matrix

Arguments

Return Value real(kind=dp), dimension(size(A, 1), size(A, 2))