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Questions tagged [numerical-methods]

Questions on numerical methods; methods for approximately solving various problems that often do not admit exact solutions. Such problems can be in various fields. Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions.

In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems.

Definitions: Numerical methods are techniques to approximate mathematical procedures (example of a mathematical procedure is an integral).

Approximations are needed because we either cannot solve the procedure analytically (example is the standard normal cumulative distribution function) or because the analytical method is intractable (example is solving a set of a thousand simultaneous linear equations for a thousand unknowns for finding forces in a truss).

Applications: With the advent of the modern high speed electronic digital computers, the numerical methods are successfully applied to study problems in mathematics, engineering, computer science and physical sciences such as biophysics, physics, atmospheric sciences and geo-sciences.

Possible topics include but are not limited to:

  1. Approximation theory, interpolations.
  2. Numerical ODE/PDE.
  3. Root finding algorithm.
  4. Numerical linear algebra, matrix computations.
  5. Discrete integral transform, FFT, etc.
  6. Linear/Non-linear programming, integer optimization.

For questions concerning matrices, please consider adding the tag.

For questions concerning optimization, please consider adding the tag.

For questions concerning Numerical ODE/PDE, please consider adding the // tag.

References:

https://en.wikipedia.org/wiki/Numerical_method

"Numerical Methods for Scientific and Engineering Computation" by M. K. Jain, S.R.K. Iyengar, R. K. Jain

14158 questions
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Finding the rate of convergence in a function already in Maclaurin series rep.

I am a little stuck when it comes to finding the rate of convergence for the function $f(x)=\frac{1-cos(x)}{x^2}$. The first part of this question was to show that the limit was $\frac{1}{2}$, and I was able to accomplish that using the Maclaurin…
CleoFroPollo
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Why is restarted GMRES used?

I know the GMRES (Generalized minimal residual method) is a method to solve a system of equations Ax=b where A is non singular matrix of size m by m. My question what is restarted GMRES method ? Why is it used?
Mike
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How to estimate $\alpha$ given a discrete sequence $a(n) = n^{\alpha}+ n^{1-\alpha}+b$.

I have a data sequence of length $10^6$ which I know it can be approximately modeled as $a(n) = n^{\alpha}+ n^{1-\alpha}+b$ ($b$ is an unknown constant, $0<\alpha <1$). But how to estimate $\alpha$ here? Update: Sorry, I made some mistakes above. I…
Jimmy Kang
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Determining interval for the square root using Heron's Algorithm.

My question is about using prescaling for Heron's Algorithm as described in on page 4 in this textbook: http://assets.press.princeton.edu/chapters/s9487.pdf I am able to understand that we are limiting our solution only to nonnegative numbers since…
Andrew Shedlock
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How do test function analysis results carry over to general equations

In numerical analysis, when considering a numerical method to solve a differential equation (think of Euler Forward, Euler Backward, Runge-Kutta 4, etc.), usually all the basic properties of the method (truncation error, stability, convergence) are…
user161518
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Determine the tangent line at $f(x_0)$

I know that the tangent is on the form $y=ax+b$ but how should I show this one: Let $f:\mathbb{R}→\mathbb{R}$ be a differentiable function. Consider a sample point $(x_0, f(x_0))$. (a) determine the tangent line at $f(x_0)$, i.e. the straight line…
user530832
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Determining Parameters of a two-step Method

I am trying to take a closer look at multi-step methods. A two-step method is given by $$ \alpha_0 y^{(j)} + \alpha_1 y^{(j+1)} + y^{(j+2)} = \tau ( \beta_0 f^{(j)} +\beta_1 f^{(j+1)} + \beta_2 f^{(j+2)} )$$ $ \alpha_2 =1 $ The aim is to…
pastapesto23
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Numerical approximation of function root using the secant method

The task is as follows: Find the biggest negative root of the following equation using the secant method: $$\tan{x}-x+\frac{1}{2}=0$$ with $\epsilon =0.02$ being the max error. So what I don't understand in this task is the meaning of the biggest…
A6EE
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What is epsilon algebra and why is it important in Numerical Analysis?

My professor is using the following slides: What is epsilon algebra and why is it important in Numerical Analysis?
user366312
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Doubt in forward difference operator in numerical Methods

To find $\triangle^3(1-2x)(1-3x)(1-4x)$ In my textbook it is given as - $P_n(x) = a_0+a_1x+a_2x^2+....+a_nx^n$ be n degree polynomial. THen $ \triangle^kP_n(x) = 0, k>n; \triangle^nP_n(x) = a_0(n!)$ But how to prove this?
Magneto
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Backward Euler's Method with Newton

The linear backward Euler's Method is given by $$ y^{j+1}=y^j+ \tau k_1 $$ $$ (I- \tau J) k_1= f(y^j), J:= f'(y^j)$$ To show, is that $y^{j+1}$ is equal to $\tilde{y^{j+1}} $ the Output of the Backward Euler's Method, if you proccess one…
pastapesto23
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Proving iterative methods to calculate a linear system of equations converge.

Given the $n\times n$ matrix $$B=\begin{bmatrix} \frac{1}{2}&\frac{1}{2^2}&\frac{1}{2^3}&\cdots &\frac{1}{2^n}\\ a &0&0 &\cdots&0 \\ 0 &a&0&\cdots &0\\ \vdots&\ddots &\ddots&\ddots&\vdots\\ 0&\cdots&\cdots&a&0\end{bmatrix}$$ I want to prove the…
mathie12
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Find the coefficients so the integration formula integrates exactly polynomials of degree $3$.

I want to find the coefficients $A_0,A_1,C_0,C_1$ so that this integration formula integrates exactly polynomials of degree $3$ or less. Prove that this formula is not exact for polynomaials of degree $4$.…
mathie12
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Doubt in derivation of Regular falsi method

I have doubt in derivation. Pls clarify Let points be $(a, f(a)), (c,0), (b,f(b))$ $y-f(a) = m (x-a) = \frac{f(b)-f(a)}{b-a}(x-a)$ Substituting y=0, x=c, $0-f(a) = \frac{f(b)-f(a)}{b-a}(c-a) \implies \fbox {c=a-f(a) ( $\frac{b-a}{f(b)-f(a)}$)}$…
Magneto
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Show that $|f''(\xi)|\leq2$ on the actual interval.

We want to compute an approximation for $\int_{-1}^1e^{-x^2} \ dx$ using trapezoid method. The trapezoid rule for an interval is $$\frac{b-a}{2}(f(a)+f(b))=\int_a^bf(x)\ dx +\frac{f''(\xi)}{12}(b-a)^3, \quad \xi\in(a,b).$$ Show that…
Parseval
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