# Questions tagged [numerical-linear-algebra]

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

2,207 questions
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### QR-algorithm complexity on a symmetric tridiagonal matrix

Why does the QR algorithm (for calculating eigenvalues) only require O(m) calculations per step when performed on a symmetric tridiagonal matrix?
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### algebra odd numbers

A question states, using algebra, prove that when the square of any odd number is divided by four, the remainder is $1$ I managed to go up to $4(n^{2}+n)+1$, from $(2n+1)^{2}$ but I dont know how to ...
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### Point intersecting line, and finding implicit equation for line given parametrisation.

Consider a point $A= (1,3,9)$ in $\mathbb{R}^3$, and a line $L$ defined by the parametric equations $$x=2+2t, y=2, \textrm{and } z=10+5t.$$ (a) Determine if $A$ lies on $L$. (b) Write a general, or ...
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### Counterexample of closure of subset under vector addition and scalar multiplication. [on hold]

Determine if the following set is a vector space over real numbers. Prove your answer. You need to check only the closure of the set under the operations. If a property is false, provide specific ...
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### How to solve a matrix dominated by zeros?

I am trying to solve a matrix of this form: Is there a known algorithm or a method to solve this kind of matrices more efficiently than a normal Gauß elimination method? I input the diagonals as ...
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### Treatment of Floating Point Rounding in Trefethen & Bau

Something I noticed in the Trefethen & Bau Numerical Linear Algebra book is that, after introducing elementary floating point arithmetic, they do not pay too much care to the initial rounding of ...
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### How to use Python to calculate the limit of a matrix exponent? [closed]

How would I go about writing a Python script to find the limit of a matrix to an exponent given M is a square matrix? $$\lim_{ n\to \infty } M^n$$ Any advice or resources would be greatly ...
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### Difficulty in understanding the program for Gaussian elimination using full pivoting

I am a second year open university BS mathematics student taking a course on numerical methods. I thought it would be good idea to implement some of these algorithms in C++ - to learn how numerical ...
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### How can I prove that an algorithm is numerically stable?

I come from Computer Science and I designed an algorithm belongs to Numerical Linear Algebra field. The analysis of algorithms in Computer Science usually involves the correctness, time and space ...
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### Constructing a degree-1 Lagrange interpolation polynomial

Construct the Lagrange polynomial $p_1$ of degree $1$ for a continuous function $f$ on $[-1, 1]$ using the points $x_0 = -1$ and $x_1 = 1$. My attempt (note: I am using the notation that is used on ...
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### Construct a Real Matrix for given Complex Eigenvalues

I need to construct real-valued matrices with specific complex eigenvalues. I have seen the companion matrix, which sort of does my job, but there are some other desirable properties as well, so I'm ...
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### Solving for $L$, where $P_n \cdots P_1 = I - XLX^T$, with $\Vert x_i \Vert_2=1$, $P_i := I - 2x_i x_i^T$, $X=[x_1 \cdots x_n]$

Problem Solve for $L$, where $P_n \cdots P_1 = I - XLX^T$ where $\Vert x_i \Vert_2=1$, $P_i := I - 2x_i x_i^T$, and $X_{m \times n} = [x_1 |\cdots | x_n]$. Try Note that $L$ is a lower ...
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### Hessenberg reduction via Householder reflector

I'm trying to understand $QR$ algorithm, and one oeﬃcient way of using the QR algorithm is to ﬁrst transform the matrix to Hessenberg form using Householder reflectors. We use the Householder ...
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### Prove that $\lim_{k\rightarrow > \infty} \frac{\|A^{k+2}x\|}{\|A^{k}x\|}=\lambda^2$

Assume that $A \in \mathbb R^{n×n}$ has $n$ linearly independent eigenvectors $u_1, u_2, . . ., u_n ∈ \mathbb C^n$ with associated eigenvalues $λ_1, λ_2, . . ., λ_n$ with $λ_1 = λ, λ_2 = −λ$, for ...
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### the reference for block tridiagonal matrix of finite element discretization of 2D convection diffusion equation.

I need to know a block tri-diagonal matrix with Kronecker product structure arising from finite element discretization of 2-D convection-diffusion equation on square domain to test some codes. But I ...
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### Show convergence of an algorithm within $m$ steps

I am trying to show that the following algorithm outputs the solution to the problem $Ax=b$. Assumptions $A$ is symmetric positive definite of size $n \times n$ with $m$ distinct eigenvalues. The ...
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### Showing $\sup_{\Vert w \Vert_\infty} |v^\ast w| = \Vert v \Vert_1$ for $v \neq 0$

Problem Show, for $v \neq 0 \in \mathbb{C}^n$ $$\sup_{\Vert w \Vert_\infty=1} |v^\ast w| = \Vert v \Vert_1$$ And find the similar equality for $\sup_{\Vert w \Vert_1=1} |v^\ast w|$ Try ...
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