Nested multiplication
How many multiplication and addition operations are needed in order to evaluate a polynomial like
$$ P(x) = a_0 + a_5x^5 + a_{10}x^{10} + a_{15}x^{15} $$and how to reduce the number of these operations?
Simple Markov chain
A exercise in a book called Introduction to linear algebra by Strang (2016) first defines a matrix of coefficients ($\mathbf{A}$), a vector of starting values ($\mathbf{u}_1$) and then asks for computing successive values $\mathbf{Au}_1 = \mathbf{u}_2$, $\mathbf{Au}_2 = \mathbf{u}_3$, $\mathbf{Au}_3 = \mathbf{u}_4$, and to explore if any interesting properties appear. Also, the exercise asks for a program that does the computation in some programming language, so let’s see how to do this in C++.
Generating random numbers in C++
As is described in Forsyth (2018), the definition of normal distribution, which is a distribution that describes a distribution of a random variable ($x$), given a mean ($\mu$) and variance ($\sigma$) as
$$ \phantom{,} \ f(x \ | \ \mu, \ \sigma) = \dfrac{1}{\sqrt{2 \pi \sigma^2}} \exp \left( - \dfrac{(\mu - x)^2}{2 \sigma^2} \right) \ . \tag{1} $$ $$ \begin{array}{c c c} \phantom{=} & f(x \ | \ \mu, \ \sigma) & \phantom{=} \\[1em] = & \dfrac{1}{\sqrt{2 \pi \sigma^2}} \exp \left( - \dfrac{(\mu - x)^2}{2 \sigma^2} \right) & \phantom{=} \end{array} \tag{1} $$ When $\mu = 0$ and $\sigma = 1$, $(1)$ can be written
1st Derivative of the Sigmoid Function
In neural networks a activation function is a function that defines a threshold that makes a node of a neural network to activate. One example of a such activation function is the sigmoid function
$$ \sigma(x) = \dfrac{1}{1 + e^{-x}}. $$(Aggarwal, 2023; Rojas, 1996).When training a neural network, activation function’s derivative is needed, but what it is for the sigmoid function?