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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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Interpolation Polynomial-Newton Method, infinite value

This is a question on my homework. Let $P_n$ be the interpolation polynomial of f at the nodes $x_0, \ldots , x_n$. Suppose we know $P_n(x) = 3x^n + Q(x)$, where $Q$ is a polynomial of degree at most $n − 1$. Find $f[x_0,\ldots , x_n]$. I am…
girlwithadragontattoo
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Exactness of Simpson's Rule of numerical integration

Given $f(x)=\exp (i 2\pi x)$. What is the minimum number of data points $(x,f(x))$ so that $\int_0^1 f(x)\,dx$ is exact when calculated with Simpson's $-1/3$ rule of numerical integration? The answer given is $9$. I have no idea how to prove it.
Purushothaman
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Modified Newton method - explanation

Could someone describe the Modified Newton method or give a reference? Method: $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_0)}$$
user156262
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Trapezoidal Rule Negative Error

Im supposed to explain the reasoning behind retaining a negative error from a trapezoidal approximation of the function: f[x] = e^4x. I have graphed the function and it looks like this: Graph of Function I think the main reason I received a negative…
Mike
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Conditions for which two matrices multiplied together can be separated using PCA

Suppose that I have two real-valued matrices $\bf{A}$ and $\bf{B}$. Both matrices are exactly the same size. I multiply both matrices together in a point-by-point fashion similar to the Matlab A .* B operation. Under what conditions can I…
Nicholas Kinar
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Precision of the root using bisection method

It is known that the root of the function $f(x)$ lies between $0$ and $1$. After $10$ iterations of the bisection method, what will be the approximate precision of the root?
Arjun Upadhyay
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What is meant by consistency for one step methods?

Does anyone know what it means for a one-step method to be 'consistent'? I've seen it written that if $a + b = 1$, then the RK2 method is consistent. How can I show that if $a + b = 1$ then RK2 is consistent? RK2 being $$\begin{aligned} y_{i+1} &=…
sonicboom
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Building newton interpolaton polynomial

Let $x_k, y_k \in \Bbb R, \ x_k = kh, \ k = 0, 1, 2$ and $h \in (0,1)$. I want to build an interpolation polynomial $p \in P_2$ (so a polynomial of degree $2$) with the help of the Newton base. Following the formula, I receive $$p(x) = y_0 + {{y_1…
Julian
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Show that $p'(x_0) = {{4y_1 - 3y_0 - y_2} \over 2h}.$

It was part of an bigger excercise to calculate the Newton polynomial $$p(x) =y_0 + {{y_1 - y_0} \over h} (x-x_0) + (-2y_0 + 4y_1 - 2y_2)(x-x_0)(x-x_1).$$ Now, I need to show that $$p'(x_0) = {{4y_1 - 3y_0 - y_2} \over 2h}.$$ But solving this by…
Julian
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Determine a quadrature formula $Q[f] := 2/3f(x) + 1/3f(y) \approx \int_0^1 f(z) dz$

Given a quadrature formula for functions $f: \Bbb R \rightarrow \Bbb R$ of the form $$Q[f] := 2/3f(x) + 1/3f(y) \approx \int_0^1 f(z) dz,$$ determine every $(x,y)^T \in \Bbb R^2$ such that the formula integrates polynomials of maximal degree…
Julian
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Determine a quadrature formula $Q[f] = w_0f(0) + w_1f(\pi/2) + w_2f(\pi)$ in order to calculate $I[f] = \int_0^\pi f(x) dx.$

Determine a quadrature formula $$Q[f] = w_0f(0) + w_1f(\pi/2) + w_2f(\pi)$$ in order to calculate $$I[f] = \int_0^\pi f(x) dx.$$ The formula shall integrate every function of the space $G = \operatorname{span} \{1, \sin x, \cos(x/2)\}$ exact. I…
Borol
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Show that $\sum_{k=0}^n x_k^r l_k(x) = x^r$

Let $n \in \Bbb N, x_k \in\Bbb R$ pair wise different grid points for $k = 0, \ ..., \ n$, and define the Lagrange polynomials $l_k(x) = \prod_{j=0, j \neq k}^n {{x - x_j} \over {x_k - x_j}}$. For $r = 0, \ ..., \ n$, Show that $\sum_{k=0}^n x_k^r…
Julian
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Solving an integral with weight function $w(x) = e^{-|x|}$

Given the weight function $w: (-\infty, \infty) \rightarrow \Bbb R$, $w(x) = e^{-|x|}$, I need to show that $\int_{-\infty}^{\infty} w(x) dx = 2$. Unfortunately, I still have a lack of understanding about what the weight function actually is. I…
Julian
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Numerical methods for finding the solution to equation

Could someone show the procedure for the following two numerical method problems: Solve an equation $f(x)=x^2-e^x+2=0$ with precision up to $10^{-4}$ using the secant method. Using the tangent method, find one positive root of equation $\cos…
user300045
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Numerical integration

Determine coefficients $A_k ; k=1,2,3,4$ in the following quadratic formula $$\int_{-1}^{1}f(x)dx=A_1f(-1)+A_2f(1)+A_3f'(-1)+A_4f(1)+R$$ such that it has the maximum possible algebraic degree of accuracy. Evaluate an error, and using the obtained…
user300045
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