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

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1answer
51 views

Induced Matrix Norm

I have trouble following a proof of the induced Norm $||\cdot||_1$ The proof can be found here: ...
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0answers
19 views

Maximal angle between kernel of rows of a matrix

Consider a matrix with 2 columns $$ \begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \\ a_3 & b_3 \\ \vdots & \vdots \end{pmatrix} . $$ To each row $(a_i \;\;\; b_i)$, one draws the kernel ...
2
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1answer
129 views

Check if the following gradient is correct

This question regards the verification of the gradient of a given function. Notation. Let $N, K \in \mathbb{N}_0$ be given (nonzero) integers, with $K > N$. Let $\mathbf{x} = [x_b \ y_b \ z_b]^T ...
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1answer
24 views

How to evaluate the accuracy for sparse linear system solver

I'm currently trying to do some experiments on linear solver. However, it's a little hard to get the sense of the numbers. For example, I know large condition number is bad, but how large is bad? ...
2
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1answer
68 views

Least squares problem with orthonormality constraints

Given $y_1,\ldots,y_n\in \mathbb{R}$,$w\in \mathbb{R}^d$, and $x_1,\ldots x_n\in \mathbb{R}^D$, how do we solve the following optimization problem \begin{align} \min_A \sum_{i=1}^n (y_i-w^TA^Tx_i)^2\\ ...
2
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1answer
284 views

equations solved with Newton's method by finding the zeros of functions?

I found this statement in one paper I read recently: This problem can be solved by finding the zero of functions: ...
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0answers
28 views

Not enough memory for GMRES

After realizing that Gauss-Seidel is extremely slow for my simulation, i wanted to try GMRES and luckily found the C++ code here without diving into the theory. The size of the matrix in my case is ...
1
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1answer
42 views

QR decomposition error

How to find $$||A - QR||_2$$ without finding Q matrix (A is matrix, QR - qr decomposition of A) I have written a code which return only R (using Householder transformation).
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2answers
472 views

Gauss-Seidel method convergence algorithm

From Wikipedia: The convergence properties of the Gauss–Seidel method are dependent on the matrix A. Namely, the procedure is known to converge if either: ...
2
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1answer
135 views

Finding vector $x$ so that $Ax=b$ using Householder reflections.

Assume $n\times m$ matrix $A$ and vector $b$ are given. I am looking for $x$ that satisfies $Ax=b$ in terms of linear least squares problem. Let $A=\begin{bmatrix} 1 & 1 & 1 \\0 & 1 & ...
1
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1answer
331 views

block matrix multiplication

If $A,B$ are $2 \times 2$ matrices of real or complex numbers, then $$AB = \left[ \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right]\cdot \left[ \begin{array}{cc} b_{11} ...
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0answers
36 views

Solving a linear equation with a moderatly sparse 1000000*1000000 symmetric matrix

I've got a linear equation $A_{n*n}\cdot x=1_n$, where $n=1000000$ and $A$ is symmetric with approx $1000$ nonzero entries in each column and row. What would be the best numerically stable algorithm ...
2
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1answer
65 views

fast multiplication for a matrix and its transpose.

I know Strassen and other methods can achieve better than $O(n^3)$ for general square matrix multiplication. I am curious of the spacial case where the multiplication is between a $n*m$ matrix $A$ ...
1
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1answer
259 views

Proof that Newton Raphson method has quadratic convergence

I've googled this and I've seen different types of proofs but they all use notations that I don't understand. But first of all, I need to understand what quadratic convergence means, I read that it ...
3
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1answer
55 views

Spectral norm of symmetric matrices only with the diagonal, the first column, and the first row non-zero

Consider a real symmetric matrix $$\mathbf{M}=\left[\array{a_0&a_1&a_2&\cdots&a_n\\ a_1&b_1&0&\cdots&0\\ a_2&0& b_2&\cdots&0\\ \vdots &\vdots ...
0
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1answer
75 views

Is this matrix guaranteed to be positive-definite?

(This question pertains to a larger model proposed in this SIGRAPH paper, but I've pulled out the pertinent question and generalized it.) Let's say I have two large matrices $M$ and $L ...
0
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1answer
26 views

Matrices in Linear Algebra

Let: $ u: R^2 --> R^3$ be defined by: $$ u(x,y)=(x+2y, 2x-y, 2x+ 3y)$$ Give the matrix $M[u]$ in the canonical base of its definition space. This question might seem sort of stupid, but it was ...
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0answers
57 views

Francis Algorithm (Implicit QR Algorithm)

In Numerical Analysis, we are touching upon QR and Francis Algorithm. I understand that for Francis's Algorithm, we reduce the matrix to its upper Hessenberg form using Householder transform. What I ...
4
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1answer
277 views

9 point stencil for Laplacian operator

Given the following 9 point Laplacian \begin{align} -\nabla^2u_{i,j} = \frac{2}{3h^2}\left[5u_{i,j} - u_{i-1,j} - u_{i+1,j} - u_{i,j-1} - u_{i,j+1} - u_{i-1,j-1} - u_{i-1,j+1} - u_{i+1,j-1} - ...
1
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1answer
37 views

QR decomposition

Assuming that we have a QR decomposition of a matrix $A \in \mathbb{R}^{m \times n}$, $Q \in O(m)$ and $R \in \mathbb{R}^{m \times n}$ such that $A=QR$, where R is an upper triangular matrix. Now ...
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1answer
39 views

proving a theorem of alternative

I've read the following exercise in my book: Let $A\in\mathbb R^{m\times n},b\in\mathbb R^m,c\in\mathbb R^n$. Then exactly one holds: $Ax=0,c^t\cdot x=1$ with $x\geq0$ has a solution $A^ty\geq c$ ...
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1answer
211 views

Trapezoid Rule to Simpson's Rule Extrapolation

I need to show that one extrapolation of the trapezoid rule leads to Simpson's rule. I've looked through the other posts on ME, specifically the post with the same title, and this for help, but I ...
2
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1answer
104 views

(A + D)x = b … efficiently!

let say we have $A$ to be symmetric positive-definite (SPD), moreover block tridiagonal Toeplitz matrix and $D$ is block diagonal SPD (both with full rank). Let say we know everything about $A$ and ...
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1answer
34 views

Its about Finding the values of (A) for which the system has no solution, infinitely many solutions, and a unique solution in linear Algebra [duplicate]

I really couldn't find the answer no matter how i tried Plz Help when I tried to solve it i got a really big numbers like a^6 ..etc' Okay what I did with this question is solving it by reducing the ...
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0answers
53 views

Upper Hessenberg Form

I am given a matrix. I would like to reduce it to its upper Hessenberg Form. We are discussing eigenvalue computations in Numerical Analysis and the textbook just gives the algorithm for it without an ...
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2answers
17 views

Is there a formula for finding the series of numbers based on an average.

So I am working on a program that does a scrolling effect. the image is 1200pixels wide, and each time it scrolls, it should move an average of 2.67 pixels. However I am using a function that does not ...
2
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1answer
38 views

Vector optimization with set constraint

This is a more generalized form of a previous unanswered question, from which I've removed all the content that wasn't relevant to the actual problem. I have a minimization problem of the form $$ ...
3
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1answer
52 views

Linear Algebra quesion

$A^{-1} - \lambda A = B^{-1} - \lambda B - \alpha v v^T$ $A, B \in S^n_+$; $v \in R^n$; $\lambda, \alpha \in R_+$. Can we solve A in term of other variables?
3
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1answer
60 views

Necessary and Sufficient conditions for convergence of matrix iterations

I need some help figuring out how to go about the iteration part of the problem...I don't really know where to start. If someone can please help take me through it that would be greatly appreciated. ...
0
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1answer
38 views

Estimating rates of convergence

If I have a set of data points obtained from a numerical approximation say 15.3828 15.2458 15.2095 15.2003 how can I estimate the rate of convergence?
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1answer
56 views

How does LU decomposition work?

I'm interested in the algorithm of LU decomposition in order to solve a LSE like $Ax=b$, where $A$ is a square matrix. My question is: When I compute $PA=LU$ do I also need to interchange rows in $L$ ...
3
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1answer
133 views

A Beautiful Determinant!

Find the determinant of the following matrix in the terms of $a_1,a_2,\cdots,a_n$ explicitly, $$ \begin{bmatrix} a_1 & a_2 & a_3 & \cdots & a_n\\ a_2 & a_3 & a_4 & \cdots ...
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0answers
44 views

Minimizing a vector constrained to a set

Sorry if this is wordy or over-complicated, I'm not sure how to isolate the problem any more than I have below without losing important context: I'm trying to implement a coordinate block descent ...
0
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1answer
3k views

Gauss Seidel iteration in matlab

I've posted this question before for crout factorization. Now, I need help with Gauss-Seidel iteration. Write a program that takes a value for n and solves for x using the following method: ...
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0answers
26 views

Simultaneous iteration of Symmetric Matrices

Given a Matrix $A$ we can use Simultaneous iteration(Using power iteration on all columns simultaneously) to compute the d biggest eigenvalues. Now this method will give you the biggest eigenvalues, ...
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0answers
32 views

Gerschgorin Theorem singularity proof

I know how to prove the Gerschgorin Theorem but how exactly would one show that there are no values of $\mu$ s.t. $\mu<0$ for which $A-\mu B$ is singular where $$ A= \begin{bmatrix} ...
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0answers
24 views

Gaussian Elimination theoretical question

You know how Gaussian Elimination can be broken up into a sequence of L-U premultiplications right? Suppose that there is a matrix $A=a_{i,j} : j=1,...,n$ is an $n × n$ real matrix such that ...
4
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1answer
49 views

QR-Decomposition of matrix valued function

I already posted the following question on MO, but id did not raise much interest there. Maybe the title is too elementary to gain research interest. Suppose I have a matrix valued function $$ ...
0
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0answers
47 views

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 ...
0
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1answer
32 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 ...
0
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1answer
58 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 ...
2
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1answer
66 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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0answers
57 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 ...
2
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1answer
39 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
152 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 ...
0
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1answer
71 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 ...
3
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1answer
99 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?
0
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2answers
34 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 ...
1
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1answer
73 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) ...
0
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1answer
90 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 ...