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

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QR Algorithm with Shifts Question

Why must QR Algorithm with Shifts make no progress when applied to this n x n matrix? (attached as image). Also, if a matrix A is orthogonal in a QR factorization, will R be tridiagonal? How would ...
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1answer
13 views

Using Givens Rotation on a vector

Say we have a vector v=$[3\ 0\ 4]$. Find a 3x3 orthogonal matrix Q such that only the second component of Qv is nonzero and such that this component is also positive. Is Q unique? I tried ...
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15 views

Inner Product inequality problem using Cauchy Schwarz, or what other way?

Let $<p,q>$ be an inner product on n. If p and q are both of degree n, show that $<p,q>^2$ $\leq$ $<p,p>$ $<q,q>$. I tried multiplying the right side out but am getting ...
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1answer
33 views

Show that the iteration $x_{n+1} = x_n - 2\frac{f(x_n)}{f'(x_n)}$ converges quadratically to $x_*$ provided $x_0$ is sufficiently close to $x_*$

We have the following conditions for the above slightly-modified Newton's method iteration: $f$ is a real function of one real variable $f''$ is Lipschitz continuous $f(x_*) = f'(x_*) = 0$ I also ...
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0answers
12 views

Odd and Even Weight functions in orthogonal polynomials proof

Suppose now that w is an even function, i.e. $w(-x)$ = $w(x)$ for all x in $[-1,1]$ and let $p_0$,..., $p_n$ be a family of orthogonal polynomials with respect to w. Prove by induction that $p_k$ is ...
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0answers
24 views

Least squares polynomial approximation $(f-p_n,q)=0$ proof.

I know how to do the other way around but I am getting stuck with showing the following If $<f-p_n,q>=0$ then $p_n$ is a polynomial of best least squares approximation in a norm $|\cdot|$ for a ...
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1answer
53 views

Non-Orthogonal Eigenvectors and Computation?

Say for a real, rectangular matrix $X$ and a s.p.s.d matrix $Q$ we maximize or minimize $Tr(X^TQX)$ under the constraint $Tr(X^TM) = 1$ for some fixed real matrix $M$. i) Would the columns of the ...
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20 views

Least squares of symmetric positive semidefinite matrices

What's the best (in terms of computation time and numerical robustness) way to find the least squares solution of $$Ax = b$$ if $A$ is symmetric and positive semi-definite? If $A$ were symmetric and ...
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1answer
25 views

Eigenvalue of (1-0) matrix

Assume I have 2 matrices, each of size nxn with only 1 and 0 as entries in both. (n>10) The first matrix (call it A) has each row summing up to 2 (ie: on each row, it has two "1" and n-2 "0"). It is ...
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2answers
57 views

Cholesky Decomposition for positive semidefinite separation

Cholesky decomposition is a common way to test positive semi definiteness of a symmetric matrix $A$. If the algorithm "goes wrong" trying to take a square root of a negative number, I know the matrix ...
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1answer
32 views

Householder matrix Uw acts as the identity on the subspace w

How can i show that a Householder matrix $U_w$ acts as the identity on the subspace $w$? and that it acts as a reflection on the one-dimensional subspace spanned by w; i.e., $U_w(x) = x$ if $x$ is ...
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1answer
40 views

Is there a faster way to calculate a pseudo-inverse of a matrix than using SVD that is as numerically stable as with SVD?

Is there a faster way to calculate a pseudo-inverse of a matrix than using SVD that is as numerically stable as using SVD?
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14 views

create sparse matrix from diagonal array in matlab

I have the 7 diagonal that come from a 3D finite volume discretization in separate arrays. These 7 arrays have 3D shape and correspond to each elements of the stencil for all points: top, bottom, ...
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2answers
23 views

Minimum of Maximum of Addition of two vectors/arrays

Suppose you have two arrays and you want to compute the maximum of the addition of the two arrays. Now you move the second array one field to the right. Now you can compute the maximum again of the ...
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1answer
37 views

Homework for Gauss Seidel method

Let A be a strictly diagonally dominant matrix. Suppose we use Gauss Seidel method to solve $Ax=b$, a sequence of vectors {$x_{0},x_{1},...,x_{k},...$} is obtained (where $x_{0}$ is the initial guess) ...
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1answer
24 views

Linear system of equations and multiple linear regression: Numerical solving

I am currently implementing a test procedure for data, namely a linear form of the Kramers-Kronig relations (paper here: http://jes.ecsdl.org/content/142/6/1885.abstract). This includes solving a ...
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14 views

cubic B-spline interpolation function

I read that the B-spline basis functions are the follows: $B_0(x)=(1-x)^3/6$ $B_1(x)=(3x^3-6x^2+4)/6$ $B_2(x)=(-3x^3+3x^2+3x+1)/6$ $B_3(x)=x^3/6$ The cubic b-spline interpolation function it ...
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0answers
50 views

convergence for symmetric, positive semi-definite operator

Assume $u$ is a vector in the Euclidean space $\mathbb{R}^N$, $||u||=\sqrt{\langle u, u\rangle}$, where $\langle u, v\rangle = \sum_{i=1}^N u_i v_i$. I have that $||u^{k+1}-u||\leq ||I - c ...
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1answer
26 views

Reference request: nonlinear systems, optimization, ode/pde

Could someone suggest me one or more good books on the following topics: Nonlinear systems: fixed point and Newton's method Optimization: steepest descent and Newton's-quasi newton methods ODE ...
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40 views

Standard symmetric tridiagonal matrix Eigenvalue decomposition algorithm?

Hi I am trying to generate an arbitrary Gauss quadrature rule by using the Golub-Welsh algorithm (here). I need to code this on C++ for my personal project. This algorithm involves the eigenvalue ...
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21 views

Diagonal Pivoting Algorithm

Commonly in LU factorization, partial pivoting is used. I know there is another pivoting which is diagonal pivoting. However, on the internet very few resources discussing diagonal pivoting (Only ...
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0answers
22 views

Numerical Linear Algebra- Proof for the Backward stability of inner Product?

I know that inner product is backward stable through various sources but to prove this statement is which I'm not aware of...if someone could help me with that that would be great. ...
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1answer
39 views

Numerical Linear Agebra

how to Prove the backward stability of the inner product ? ...
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2answers
51 views

Exponential of a 3x3 lower bidiagonal matrix

I have a 3x3 matrix with non-zero entries ONLY along the main diagonal and the diagonal above. There are exactly two non zero diagonals in the matrix like this \begin{pmatrix} a & 0 & 0 \\ d ...
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1answer
29 views

How to make a function lie in the interval [0,1]

Is there a way to convert a function g(x) so that the result lies between [0,1]? Thank you in advance.
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1answer
69 views

Reducing a matrix to upper Hessenberg form using Householder transformations in Matlab

I have the below Matlab code based on what my professor gave me in class. The last line of this code is giving me an incompatible dimensions error. When I checked the size, I got that the matrix ...
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1answer
17 views

Solving Ax = b where A is composed of diagonal blocks

I would like to solve the equation $Ax=b$ where $x\in\mathbb{R}^n$ and $A$ is of the form: $$A= \begin{bmatrix} D_1 & D_2 &D_3 \\ D_2 & D_4 & D_5 \\ D_3 & D_5 & D_6 ...
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1answer
16 views

Checksum Invariants for Matrix Inversion via Gaussian Elimination

In general, when solving $Ax=b$, we make the $[A|b]$ matrix and doing row operations to reduce the left hand side to an identity. It's painfully annoying to find mistakes in the process. Assuming we ...
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0answers
16 views

Iteratively solve linear equations with rank-1 updates on LHS and RHS

What is the best way to iteratively solve updating equations of the form $$ Ax=b $$ $$ (A+c_1v_1^\intercal)x_1=b+ \alpha_1 d_1 $$ $$ (A+c_1v_1^\intercal+c_2v_2^\intercal)x_2=b+\alpha_1d_1+\alpha_2d_2 ...
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1answer
14 views

Equality of iterates produced by Minres and GMres for practically symmetric matrix

My system is from time-integration of the semi-discretized Stokes equation. The time update of the variables $(v,p)$ is defined via the solution of $$ \begin{bmatrix} A & -\tau B^T \\ B & 0 ...
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36 views

Examples of non trivial problems in this structure.

I'm looking for examples of non trivial problems that match with the follow structure. Let the function $$g: U \times V \rightarrow \mathbb{R}$$, where $U$ and $V$ are complex vetorial spaces of ...
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1answer
33 views

Legendre Polynomial Orthogonality and Size

Show $(P_i,P_j)=\begin{cases} 0& i \neq j \\ \frac{2}{2j+1} & i = j\end{cases}$ for $0 \leq i, j\leq2$ I'm just not sure exactly what I'm supposed to do. Do I plug in values of i and j and ...
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1answer
31 views

Householder Reflection

I am working on algorithms for SVD by first performing Householder transformation. I got my algorithm to work but I'm trying to gain a better intuition of it. My understanding is that the ...
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1answer
27 views

Can a tridiagonal matrix be rectangular?

My program works with tridiagonal matrices (calculates its LU decomposition) so before doing anythig, it stores the matrix in 3 vectors: the three diagonals only. So far my conclusion was, a ...
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0answers
26 views

Rank and solvability of a matrix

I am working with linear complementarity problems (LCPs) which look for a solution $\mathbf{x} \in \mathbb{R}^{n}$ in the form $$ \begin{matrix} \mathbf{x} & \geq & 0 \\ ...
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2answers
32 views

Square Idempotent matrix: efficient algorithms for finding eigenvectors

Given a square idempotent $N \times N$ matrix $A$ with large $N$, and a priori knowledge of the rank $K$, what is the most efficient way to compute the $K$ eigenvectors corresponding to the $K$ ...
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1answer
31 views

matrix exponential and Spectral abscissa

Prove that $\lim_{t \rightarrow \infty} \|e^{tA}\| = 0$ if and only if $\alpha(A) < 0 $, where $\alpha$ is the Spectral abscissa, defined as $\max{Re(\lambda_i)}$. I tried to approach this ...
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1answer
28 views

Can SVD help to solve (inequality) constrained least squares problem?

Consider the following minimization problem: $$ ||Q u - h^{o} ||^{2} \to min \;\;\; s.t. \; u \geq 0 $$ where $Q$ is $m \times n$ matrix and $u$ is $n$-dimensional vector and $h^{0}$ is ...
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2answers
27 views

non-sequential sequence function

if i remember correctly (i had one workshop on numerics years ago, sorry for my lack of knowledge) there is a way to create some sort of hash function that gives you a non sequential sequence. This ...
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0answers
29 views

Smallest set of Liner equations, which exactly fit a set of points

I have a set of 2-d points,(it can be of any arbitrary dimension n). I want to find the minimum set of straight lines(linear equations) which exactly passes through the given 2-d points (unlike ...
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4answers
120 views

Finding nonnegative solutions to an underdetermined linear system

Here's the environment of my problem: I have a linear system of 4 equations in 8 unknowns (i.e. $Ax = b$, where $A$ is $4 \times 8$, $x$ is $8 \times 1$, and $b$ is $4 \times 1$, with $A$ given and ...
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1answer
48 views

Is there a limit for how “good” a numerical method can be?

Multiplying two matrices $A \cdot B$ of size $n \times n$ in the trivial way requires $n^3$ computations. However, more efficient algorithms such as the Strassen algorithm have a lower complexity of ...
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3answers
52 views

About matrix products $A^{T}A$ and $ AA^{T} $

I'm investigating the relationship between 2-norms and eigenvalues of $A^{T}A$ and $ AA^{T} $, in order to better understand the SVD decomposition. How can I prove that $A^{T}A$ and $ AA^{T} $ are ...
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0answers
40 views

Inverse of the sum of a symmetric and diagonal matrices

I have two square matrices, $A$ and $B$. $A$ is a block symmetric matrix with 1's along the diagonal (and therefore 1's in whole blocks along the diagonal). $B$ is a block diagonal, with the same ...
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0answers
8 views

need to determine weights so that quadrature formula holds

Let $l$ be an interval on the real axis, $t_1,...,t_n$ be distinct $n$ points, then there exists n numbers $m_1,...,m_n$ such that the quadrature formula, $\int_l p(t)dt = m_1p(t_1) + ... + ...
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1answer
19 views

Complexity of sparse back substitution

What is the complexity of sparse backsubstitution $Rx = b$, given $n$, the dimensions of dense $x$ and $b$ as well as of the sparse $R$ and $nnz$, the number of nonzero entries in $R$?
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1answer
33 views

How to solve an Optimization problem with linear as well as Quadratic constraints.

I want to solve the following problem, \begin{equation} \begin{aligned} & \underset{\mathbf{x}}{\text{minimize}} & & \mathbf{x^T}\mathbf{Px} \\ & \text{subject to} & & ...
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0answers
41 views

Expressing rank condition of a matrix in terms of its elements

Let $x \in \mathbb{R}^{n}$, define $X = xx^{T}$. I have an optimization problem with some linear constraints and few quadratic constraints, and I have to solve for $x$. Using $X$ as the unknown ...
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2answers
367 views

Algorithm to find an orthogonal basis (orthogonal to a given vector)

Let $K$ be a given integer, with $K$ even (and "large"). Let $\mathbf{v} \in \mathbb{R}^{K \times 1}$ be a given non-zero (column) vector. Write a (possibly efficient) algorithm to construct a matrix ...
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1answer
32 views

Is it possible to solve a system of equations comprising FFTs?

Consider the following known matrices, A, B, C and these unknown matrices X,Y, all of which comprise values in the Real domain. Also consider $F(x)$ as the *Fast Fourier Transform function* (the ...