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Mathematics Stack Exchange

Questions tagged [numerical-linear-algebra]

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

Questions tagged with this tag can be about, but not limited to:

  1. Matrix decompositions like SVD, QR, Cholesky, etc.
  2. The solution of linear systems and least squares problems.
  3. Analysis of numerical linear algebra algorithms like condition numbers and stability analysis.
  4. Eigenvalue problems.
  5. The designs of direct or iterative methods to solve linear systems.
3541 questions
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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 matrix. Which $\rho$-norm minimizes $A$'s condition…
Hila
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1 answer

How to prove that the inner product is positive unless $Ax = b$?

Suppose $Ax =b$, then the equation above = $0$ Spp $Ax \neq b$, since $A$ is positive definite, then Am I going to the right direction for this proof? How can I show the rest is positive as well?
shimura
  • 117
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Dense Rank Deficient Linear System

What are some of the best methods for solving a Dense Rank Deficient Linear System $Ax = b$, where $A$ is Dense, Symmetric but possibly Rank Deficient. I know SVD can solve it pretty nicely while constraining the solution to be unit which is…
Zahid Hossain
  • 21
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Does the Conjugate Gradient Method provide an eigenvalue estimate?

Suppose that we apply a Krylov subspace method to the linear system $A x = b$. For example, if $A$ is symmetric positive-definite, then the Conjugate Gradient method may be used. I remember that the iterations of Krylov subspace methods also reveal…
shuhalo
  • 7,485
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Numerically stable way to compute $\text{Trace}[\mathbf A\mathbf A_1^{-1}\mathbf B\mathbf B_1^{-1}]$

I have two dense (column-major) PSD matrices $\mathbf A$ and $\mathbf B$, $\mathbf A,\mathbf B\in\mathbb R^{n \times n}$ ($n$ is usually $\sim 1000$) and also $\mathbf A_1=\mathbf A+\eta\mathbf I_n$ and $\mathbf B_1=\mathbf B+\eta\mathbf I_n$ (for…
Soumyajit De
  • 111
1
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1 answer

Question regarding Givens Rotation

I need to solve the following equation using Givens Rotation: $$ A\cdot x = b $$ Correction: I need to solve: $$ ||A\cdot x - b || \to \min $$ with $$A = \begin{bmatrix} 1 & 1 \\ -2 & -7 \\ 0 & -5 \\ \end{bmatrix}$$ and $$b= \begin{bmatrix} 1…
Angelo.Hannes
  • 281
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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).
ltc985
  • 43
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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 to handle linear equations of that size?
Michael
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Ill-conditioned matrix

Possible Duplicate: Inverse matrices are close iff matrices are close Consider this problem: $Ax = b$ I want to solve it/find x and the matrix A is ill-conditioned. Why is the fact "A is ill-conditioned" a "bad" thing?
user12358
  • 289
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1 answer

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 \end{bmatrix} \begin{bmatrix} v \\p \end{bmatrix} = f…
Jan
  • 1,008
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0 answers

The least-squares solution for interval data

I would like to solve the least-squares for $\mathbf{Ax} = \mathbf{y}$ with some values in $\mathbf{A}$ and also in $\mathbf{y}$ are interval-valued numbers. In a more detail, e.g.,: $$ \begin{bmatrix} a_{11} & a_{12} \\ a_{21} &…
zhidayat
  • 11
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How to solve a divergent linear system using iterative methods?

I have a matrix A which is symmetric and non-diagonal dominant. I tried to use Jacobi/Gauss-Seidel/SOR to solve it but it diverges. Is there any mechanism to condition the matrix for convergence using the same techniques?
codepk
  • 121
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1 answer

numerical computation without explicitly calculating certain matrices

I have to numerically multiply: $A^{-1} B A$ where B is a diagonal square matrix, and A is symmetric. A is calculated from multiplying two non-square matrices, $A = XX^T$ I know B and X, and A and its inverse are too large to be computed. X is not…
not_even_wrong
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Matrix computing

Given an $n$-vector $x$, show that floating-point computation of the Householder vector $v$ such that $P x = (I − 2vv^{T} )x = \pm\left\|x\right\|_{2}e_{1} $ gives a forward stable result $v^{\prime}$ satisfying $\left\|v^{\prime} − v\right\|_{2}…
user97011
  • 39
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Finding Matrix of Minimum 2-norm to obtain singular matrix

Let $A,X \in \mathbb{R}^{m \times m}$ with rank$(A) = m$. Find $X$ of minimum 2-norm for $A + X$ to be singular. My thoughts: Since we want $A + X$ to be singular, we want to show that it has a zero determinant (or show that it has a zero…
math student 101
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