# Tagged Questions

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

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### Response Matrix with finit actuator

I have a system of penalties ($P$) and actuators($A$). Whereby: d$P_i/$d$A_j$ = close to constant $\quad\forall i,j$ In order to minimize $P$, I create a response Matrix ($M$). With its ...
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### Looking for references on the complexity of computation of a basis transformation matrix

I'm looking for some references on the complexity for the following kind of problem: Given two Basis $(a_1, ... ,a_n)$ and $(b_1, ..., b_n)$ of the $K(x)$-vector space $V$ I want to compute the ...
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### Speeding up Conjugate Gradients iterations for Sparse Matrices?

I've been using Conjugate Gradients to minimize linear systems involving sparse matrices. Although many of my sparse matrices are highly specialized - i.e. for any given row it is easy to know which ...
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### Finite difference: Radial symmetry boundary condition in tridiagonal system?

I am putting together an axisymmetric finite difference solver for Poisson's equation over a non-"rectangular" boundary in axisymmetric cylindrical coordinates. I was planning on using the dynamic ...
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### Error bounds for solution of system of linear equations when coefficients are uncertain

I have a square system $Ax=b$ and would like to know how much the solution $x$ can change when I change the coefficient matrix $A$. I've stumbled upon the condition number, but this seems to apply ...
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### What mathematics topics pertain more towards applied mathematics?

I'm entering my second year of undergrad (majoring in mathematics), and I've found that I am really bad at Linear Algebra, but very good at Calculus and Differential Equations. I'm hoping to venture ...
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### solve nonlinear system of equation numerically

solve the following system of equations numerically $$2x+2y - e^{xy} = 0$$ $$x^3 + y - xy^3 = 1$$ I'm also asked to solve analytically but I'm pretty sure the closed form solution doesn't exist ...
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### How can I write $\prod\limits_{i = 0}^n {\left( {x - {x_i}} \right)}$ in terms of a polynomial as $\sum\limits_{i = 1}^n {{a_i}{x^n}}$?

How can I write $\prod\limits_{i = 0}^n {\left( {x - {x_i}} \right)}$ in terms of a polynomial as $\sum\limits_{i = 1}^n {{a_i}{x^n}}$? In the other words, is there a way to write $a_i$ in terms of ...
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### Most stable algorithm to solve a system of linear equations?

I am doing some image processing involving solving a system of linear equations. I am getting some errors and bits of the image looks corrupted. I would like to know what is the most stable way to ...
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### Strange behavior with coordinate transformation of square and quadrilateral

I am trying to map coordinates from a quadrilateral to a square. The coordinates are square: $(500,900)(599,900)(599,999)(500,999)$ quad: $(454,945)(558,951)(598,999)(499,999)$ where the $i^{th}$ ...
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### Notation regarding linearization near equilibrium point of dynamical system

Suppose we have $\frac{dx}{dt} = \dot{x} = f(x)$ with equilibrium point $x_e$ such that $f(x_e) = 0$. Then for the linearized approximation of the differential equation near $x_e$ we hope to use the ...
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### How to determine positive or negative definite of a bordered Hessian ?

I want to determine the minimization result I get using Lagrange Multiplier method is a local minimum by determining whether the Bordered Hessian is positive definite or negative definite.(Hopefully ...
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### Bound on the difference of matrix diagonals

I have two diagonal matrices $\Lambda,\hat{\Lambda}\in\mathbb{R}^{n\times n}$ with non-negative diagonal elements. And I have two matrices $W,\hat{W}\in\mathbb{R}^{m\times n}$, with $m\geq n$, each ...
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### maths and maths oriental doubt [duplicate]

What is 0 power 0 ? And explain verifyly.,
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### What is the source of the error in the Sherman-Morrison formula application?

The Sherman-Morrison formula results in small errors in relation to the standard matrix inverse operation after each application, as shown here. From what I understand, the Sherman-Morrison identity ...
Suppose that $A \in \mathbb{R}^{n \times n}$ is a symmetric positive semi-definite matrix such that $\operatorname{Tr}(A)\le n$. I want a lower bound on the following quantity $$\operatorname{Tr} ... 1answer 25 views ### Product rule in discrete derivative in finite difference scheme. Suppose we are on real line and I want to discretize the usual derivative operator. Take a smooth function u and step size h. Then I could define$$ \Delta_+u(i) = \frac{u(i+1)-u(i)}{h} $$as the ... 2answers 40 views ### Linear combination of basis function in logarithm space. Is it possible? I have a function f(x). As theory said that it can represent by linear combination of basis functions such as$$f(x)=\sum_{i=1}^{N}\alpha_ig_i(x)$$where \alpha is coefficient and g(.) is basis ... 0answers 15 views ### Conditional expectation of a set of Gaussian variables I was wondering if there is an efficient way to compute the conditional expectation of every element in a Gaussian random vector ? Specifically: For a pair of Gaussian random variables [x,y], the ... 1answer 29 views ### Gauß-Newton Example with one variable$$T=f(t):=2 \alpha + \sqrt{\alpha^2+t^2}$$To estimate \alpha we got the measured values T_i for t_i. Formulate the curve fitting problem and show each step in the Gauss-Newton algorithm. My ... 3answers 57 views ### Is this a circulant matrix? It's symmetric, but I'm not sure whether it is circulant. In a question that I had asked on MSE a couple of weeks ago, several commenters had said that this is a circulant matrix, and to study the ... 1answer 38 views ### Choose \rho such that \rho-norm minimizes the matrix condition number I'm solving questions from am exam that I failed miserably, so I would love it if someone can take a look at my proof and make sure I'm not making any gross mistakes. Question Let A a symmetric ... 1answer 32 views ### Automatic way to have a good initial guess for the iterative methods ( newton method) and for high dimensional nonlinear problems I am solving a variety class of nonlinear systems where I need to reduce in optimal manner the number of iterations of either the newton or modified newton method. For this end I am trying to figure ... 0answers 33 views ### How to calculate det(X^T X) efficiently, update one column of X each time X_{1} = (A, b), where X_{1} is a n\times p matrix, A is a n\times (p-1) and b is n\times1. First calculate \det(X_{1}^T X_{1}), then update b with c, st. X_{2} = (A, c) and ... 1answer 59 views ### How to numerically solve the Poisson equation given Neumann boundary conditions? I want to solve the Poisson equation on a 2D domain given Neumann-type boundary conditions: The PDE: $$\nabla^2 \; u(r,\theta) \;=\; f(r,\theta)$$ The boundary ... 0answers 67 views ### How to solve linear system of form (A \otimes B + C^{T}C)x = b when A \otimes B is too large to compute? For the given linear system:$$(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 ... 1answer 24 views ### 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 ... 2answers 31 views ### 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: ... 0answers 29 views ### Efficient method to compute grand sum of a Vandermonde matrix? Is there a computationally efficient method to calculate the sum of all elements (grand sum) of a Vandermonde matrix? Each row can be quickly calculated using the formula for a geometric progression. ... 0answers 52 views ### 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 ... 0answers 16 views ### 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} ... 2answers 39 views ### 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 ... 0answers 26 views ### 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 ... 3answers 76 views ### 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 ... 0answers 11 views ### 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 ... 1answer 48 views ### 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 ... 2answers 35 views ### 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 ... 1answer 34 views ### 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} $$where |f(x)''\leq K| and n is the number of time steps. if second ... 0answers 19 views ### Approximation of Mahalanobis distance If A is a symmetric positive definite n\times n matrix then the square Mahalanobis norm of a vector v\in \mathbb{R}^n is given by$$\lVert v \rVert_A^2=v^t A^{-1} v. Now I have a situation ...
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 ...