# Tagged Questions

Questions on the various algorithms used in linear algebra computations (matrix computations).

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### Gauss Seidel Iteration for a specific matrix

We seek to solve $Au= f$ via iteration, where $$A = \left ( \begin{array}{cc} I & S \\ -S^T & I \end{array} \right )$$ Where $S$ is an arbitrary square matrix in $R^n$ and $I$ is the ...
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### Product rule in discrete derivative in finite difference scheme.

Suppose we are on real line and I want to discretize the usual derivative operator. Take a smooth function $u$ and step size $h$. Then I could define $$\Delta_+u(i) = \frac{u(i+1)-u(i)}{h}$$ as the ...
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### Linear combination of basis function in logarithm space. Is it possible?

I have a function $f(x)$. As theory said that it can represent by linear combination of basis functions such as $$f(x)=\sum_{i=1}^{N}\alpha_ig_i(x)$$ where $\alpha$ is coefficient and $g(.)$ is basis ...
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### Conditional expectation of a set of Gaussian variables

I was wondering if there is an efficient way to compute the conditional expectation of every element in a Gaussian random vector ? Specifically: For a pair of Gaussian random variables $[x,y]$, the ...
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### Gauß-Newton Example with one variable

$$T=f(t):=2 \alpha + \sqrt{\alpha^2+t^2}$$ To estimate $\alpha$ we got the measured values $T_i$ for $t_i$. Formulate the curve fitting problem and show each step in the Gauss-Newton algorithm. My ...
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### Is this a circulant matrix?

It's symmetric, but I'm not sure whether it is circulant. In a question that I had asked on MSE a couple of weeks ago, several commenters had said that this is a circulant matrix, and to study the ...
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### Choose $\rho$ such that $\rho$-norm minimizes the matrix condition number

I'm solving questions from am exam that I failed miserably, so I would love it if someone can take a look at my proof and make sure I'm not making any gross mistakes. Question Let $A$ a symmetric ...
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### Automatic way to have a good initial guess for the iterative methods ( newton method) and for high dimensional nonlinear problems

I am solving a variety class of nonlinear systems where I need to reduce in optimal manner the number of iterations of either the newton or modified newton method. For this end I am trying to figure ...
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### How to numerically solve the Poisson equation given Neumann boundary conditions?

I want to solve the Poisson equation on a 2D domain given Neumann-type boundary conditions: The PDE: $$\nabla^2 \; u(r,\theta) \;=\; f(r,\theta)$$ The boundary conditions:...
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### How to solve linear system of form $(A \otimes B + C^{T}C)x = b$ when $A \otimes B$ is too large to compute?

For the given linear system: $$(A \otimes B + C^{T}C)x = b$$ where $\otimes$ is the Kronecker product, $A$ and $B$ are dense and symmetric positive-definite, and $C^{T}C$ is a sparse symmetric block ...
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### Can adding one column to a matrix increase its rank by more than one

Knowing the answer to this question would help me answer the following question: $A$ is an $m\times n$ matrix with $m>n$, and let $A=\hat{Q}\hat{R}$ be a reduced QR factorization. Suppose $\hat{R}$...
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### Rewrite matrix equation as a quadratic programming problem

Given real-matrix $X_{n\times p}$ how can the problem of minimizing $Tr(X^TA_{n\times n}X)$ under the constraint $Tr(X^TC)=\phi$ be posed as a standard convex quadratic program given by the form: ...
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### Fastest way to find linearly independent columns of a matrix

Given a rectangular matrix $X$ of size $n\times m$ with $m>n$, what is the fastest way to find the linearly independent coloums. Robust methods like SVD or RRQR decompostion have complexity of ...
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### Evaluating a quadratic form with an inverse of a sparse PD matrix, comparison between using the inverse vs using a Cholseky decomposition

I have the following quadratic form I need to evaluate: $x^T A^{-1} y$, where $A$ is a sparse positive definite matrix, $x, y$ are sparse vectors. Now assume that I am given for free both $A^{-1}$ ...
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### Differential equation for finding closest point on surface.

Inspired by this question I got to think about a more general case. Say I have any discretized surface and want to find closest point from each point outside of surface to the surface. Say that I can ...
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### How to apply Runge-Kutta to an implicit scheme?

I see there are some differences in the solution as I increase the resolution of my grid. I'm using Operator Splitting to solve Diffusion Reaction equation \frac{\partial u}{\...
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### Symmetry Of Differentiation Matrix

I have a problem computing numerically the eigenvalues of Laplace-Beltrami operator. I use meshfree Radial Basis Functions (RBF) approach to construct differentiation matrix $D$. Testing my code on ...
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### Relationship between QR and LU factorization

Both algorithm return very similar results in terms of having a upper/right triangular matrix as one of the factors. What is the relationship between Q and L, and between R and U? What is the ...
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### The MU-puzzle from GEB

The MUI system only uses the three letters M,U,and I to make strings. The system has four rules that allow you to make new strings out of existing strings by manipulating them. Rules 1 and 2 lengthen ...
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### How to solve the system of equations $\{10^{-4}x_1+x_2=1, x_1+x_2=2\}$ using finite precision arithmetic with three significant figures?

Consider the following two equations: $10^{-4}x_1+x_2=1$ $x_1+x_2=2$ Solve using Gaussian Elimination using finite precision arithmetic with three significant figures. I'm a little stuck ...
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### Error bound of midpoint rules with unbounded second derivative

It is well known that error bound of midpoint rule for function $f[a,b]$ is given by $$E\leq K\frac{(b-a)^3}{24 n^2}$$ where $|f(x)''\leq K|$ and $n$ is the number of time steps. if second ...
How do you prove that QR factorization via Householder Triangularization is backward stable? Theorem 16.1 (From Trefethen and Bau): Let the $QR$ factorization of a matrix $A$ be computed by ...