Referring to methods for solving linear systems of equations.

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Large binary programming problem

I have 10000 variables (each of them is binary), vector of positive coefficients and a matrix A (10000*10000), if Aij is 1, then ith and jth variables can take 1 simultaneously, if it's 0, then it's ...
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1answer
31 views

Least squares fitting

I have the following equation I came across which was solved using least squares $x = \sum_{n=1}^{N} A_{n} y_{n}$ Where $x$ is a $m \times p$ matrix and $y$ would be of size $m \times p$ as well ...
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3answers
92 views

Efficient solver for a symmetric tridiagonal system where the upper/lower diagonals are offset

I'm looking for an efficient way to solve a symmetric tridiagonal system $Mx = d$, where the upper and lower diagonals of $M$ are offset from the main diagonal by $k$ rows/columns: $$ \begin{bmatrix} ...
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1answer
55 views

Finding Interior eigenvalues using Davidson algorithm

Is it possible to find interior eigenvalues closer to some lambda using Davidson method. I was searching online but found that most people use Jacobi-Davidson method for that. Thanks
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48 views

Wanted: sequences of linear systems for recycling Krylov solver analysis

In the solution of sequences of linear systems $$A_ix_i=b_i\quad\text{for}\quad i=1,2,\dots$$ with Krylov subspace methods, data can be recycled from already solved linear systems in order to speed up ...
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47 views

Adaptive Mesh vs Uniform Mesh for multiple source/boundary/initial Data

I'm going to ask some beginners' questions. Adaptive mesh can save many DOFs than a uniform mesh. But it also needs to solve linear systems changing with mesh adaptive process. Is this not problem? ...
3
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2answers
105 views

Solving Newton-Raphson step with ill-conditioned sparse matrix

I am trying to build a complex simulator for the transport of mass & heat in porous media. I am currently following coarsely the algorithm laid out in an older software and have got the simulator ...
3
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1answer
87 views

Updating an approximate solution to a linear system in response to a small change

This question was original posted on SO but it was suggested that I post it here. I'm working on a program in which I have a banded matrix M and a vector b, and I want to maintain an approximate ...
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1answer
90 views

Problem with convergence of Jacobi iterative algorithm

I'm dealing with Jacobi iterative method for solving sparse system of linear equations. For small matrices it works well and gives right answers even if matrix is not strictly diagonal dominant, ...
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1answer
28 views

How to use TrangularView class in Eigen C++

For a nonsingular lower or upper triangular square matrix $A$, how to solve such linear system in Eigen: $$A x = b$$
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1answer
104 views

Question about extending Tikhonov regularization

I know that the Tikhonov regularization of a linear system has an analytical solution given by: \begin{equation} \hat{\mathbf{x}} = \mathrm{arg\;min}\left( \left| \mathbf{Ax} - \mathbf{b} \right|^{2} ...
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2answers
76 views

Advice on the regularisation of a linear problem

I'm numerically inverting an integral transform using a method suggested by a scicomp user from an earlier question. The problem is as follows: I wish to estimate $f(x)$ for a given $F(y)$, both of ...
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1answer
81 views

full rank update to cholesky decomposition

Let $A$ be a real, symmetric, positive definite matrix. It has at least 500 rows, possibly much more. I compute its Cholesky decomposition, which allows me to calculate $det(A)$ $A^{-1}X$ for some ...
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1answer
69 views

Implicit Finite difference scheme for a PDE with only one boundary

I am looking at a few reaction-diffusion equations of the form $\frac{dP}{dt} = D\left(\frac{d^2P}{dr^2} + \frac{2}{r}\frac{dP}{dr}\right) - a(P)$ I know the initial conditions and the boundary ...
2
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3answers
131 views

Sparse, underdetermined system of linear equations

I'm looking for an algorithm to solve the underdetermined system of linear equations $$\mathbf{A}\,\mathbf{x} = \mathbf{b}$$ with $\mathbf{A} \in \mathbb{R}^{n\times n}$, $\mathbf{b} \in ...
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19 views

Differences between methods for solving linear equation system [duplicate]

I have a huge linear equation system in this form: F=K.Δ as usual form of problems in the finite element method, where the F vector and K are known and Δ vector is unknown. There are several methods ...
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1answer
70 views

How to re-use the coefficient matrix decomposition result when solving linear systems by Eigen C++

My problem needs to solve dense symmetric linear systems something like: A x = b, A y = x, ...
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1answer
35 views

What is the fomula of polynomial time of solving positive definite symmetric linear system

For a positive definite symmetric linear system, Cholesky decomposition based method should be the best solver which has a rough n^3/3 flops requirement. What is ...
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1answer
87 views

Constrained System / Combinatorial Problem

Let $x\in \mathbb{R}^{n}$, $Y\in \mathbb{R}^{mxn}$. We can then define: $row_{i}(Y)=$ $i^{th}$ row of $Y$ $column_{i}(Y)=$ $i^{th}$ column of $Y$ $x_{i}=i^{th}$ element of $x$ $sum(x)=$ sum of ...
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1answer
81 views

indirect method for least-squares with inequality constraints

I aim to find $x \in \mathbb{R}^n$ that $\min_x |D \cdot F \cdot x|^2$ subject to $x_i = X_i$ and $x_j \geq X_j$ , $i \in I, j \in J$ and I and J partition ${1\cdots N}$ into two sets. it is ...
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1answer
137 views

Solving for null space of a matrix with mkl LAPACK

I want to find a solution for $xA=0$, where $A$ is a square matrix. I know that most of the LAPACK routines solve for $Ax=b$. So I take $A^T$ as a, and set $b=0$. I have an additional restriction of ...
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1answer
216 views

Solving system of linear equations with cyclic tridiagonal matrix

I have this problem in my textbook: Suggest efficient algorithm for solving system of linear equations with cyclic three-diagonal matrix, that is of the form: \begin{bmatrix} ...
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1answer
117 views

Sparse linear solvers in C?

I'm working on translating a discontinuous Galerkin code from MATLAB to C and I'm at the final point where I need to solve a sparse system. I've taken a course in C before but I'm very rusty and ...
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1answer
80 views

Would recalculating the residual in the conjugate gradient method help?

The conjugate gradient method suffers from an accumulation of errors as it continues. For this reason it is unwise to use it as a direct solver. My question is, would it help to recalculate the ...
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98 views

Determine the step size in a differential equation numerical solver

How can we define the precision we require in a numerical differential equation solver? What is it that I have to optimize to know? And how do I know that I'm at a sufficient time-step value? For ...
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2answers
196 views

openmp update matrix from neighboring elements (parallelise preconditioner)

Issue of data dependency with stencil code... How to parallelise this using openmp? I looked at the openmp manual, I figured out how to use DO ORDERED to get the same result as the serial version, ...
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3answers
135 views

Preconditioner for finding the smallest eigenpairs of a large, but structured, matrix

I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $L$. $L$ is a graph Laplacian, with the following structure: $L = D ...
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3answers
106 views

Pseudo-inverse of a discretized operator with a null space?

Is there a way to understand what happens when a singular operator is discretized and inverted using the pseudoinverse (say using the SVD Moore-Penrose pseudoinverse)? For example, if we discretize ...
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78 views

Lapack++ for QR algorithm

I have recently started using Lapack++ which I found convenient for my programming purpose, in general. Now, I need to solve a matrix using QR algorithm. I've searched the user manual and I found a ...
3
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3answers
124 views

Smoothly varying dense matrices arising from computational science

I have written an algorithm to solve a dense system with smoothly varying entries. This means I assume there is no large jump from any entry to its neighbors. I would love to use ...
5
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1answer
125 views

What Linear Equation Solver should be used for a problem with many dirichlet conditions?

I am solving a laplace equation on a finite-element mesh (tetrahedral, triagonal) and have many say 99% dirichlet conditions compared to the number of unknowns. Is there an efficient way to solve this ...
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3answers
252 views

What is the best solver for solving a large sparse indefinite system

What's the best solver that can solve a large sparse but indefinite matrix?
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160 views

Memory management for solving large sparse systems with UMFPACK

I'm using umfpack for solving large systems of equation. However, I'm constantly getting out of memory issues for even modest size problems as pre2, torso3, ohne2 Hamrle3 (all from Tim Devis's ...
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2answers
293 views

What is linsolve() doing?

I'm converting a piece of matlab code to python but I am having difficulty recreating the solution it finds for an overdetermined sparse matrix. The original code used the overloaded left division ...
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1answer
156 views

Advice on solving a coupled physics problem

I am taking a shot at solving a coupled physics problem. I have this matrix formed: $\mathbf{J}=\begin{bmatrix} \mathbf{A} & \mathbf{B}\\ \mathbf{C} & \mathbf{D} \end{bmatrix}$ where ...
2
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1answer
177 views

Update QR decomposition when one column is exchanged

I have got an input series of matrices $A_1, A_2, A_3, \dots $ and the difference between $A_i$ and $A_{i+1}$ is the replacement of one single column. Before i get to know $A_{i+1}$, I have to ...
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60 views

A question on CHOLMOD

When I change "cholmod_*" to "cholmod_l_" (because the size of my matrix is large, use "cholmod_" will outputs error"problem too large"), it shows " sparse:error: integer and real must match the ...
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1answer
147 views

What kinds of size of matrix A that CHOLMOD can solve Ax=b

CHOLMOD is very fast, but I am just wondering what kinds of size A such that it can solve Ax=b. I have a A of 200,000 * 200,000, but it outputs errors" problem too large". I am very appreciated if ...
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1answer
116 views

Sensitivity analysis of linear program with coin-or clp

I have written a short example to run the simplex algorithm with coin-or Clp, something quite simple like this: ...
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1answer
58 views

How to do transpose for trtrs (or tptrs) in blas?

How to do transpose for trtrs (or tptrs) in blas? I want to solve: XA = B But it seems that trtrs only lets me solve: ...
2
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2answers
219 views

Sources to get source codes for sparse matrix solvers (non-symmetric matrix)

For an implicit scheme I want to solve system $Ax=B$, where $A$ is a non-symmetric square matrix. I want source codes of large sparse matrix solvers (e.g. LU-SGS) to use in my code which is in C ...
4
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1answer
332 views

Solving linear systems by fft

I read in a paper and also at wiki that we can solve the system $$Ax=B$$ by Fast Fourier Transform, where $A$ is a circulant matrix. The solution is ...
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65 views

Why do I get “estimated error” -1.#IND when doing BICGSTAB linear solver using ILUT perconditioner in eigen

I'm using Eigen (a C++ library for numerical linear algebra) to solve a linear equation with the the bi-conjugate gradient BICGSTAB algorithm with Incomplete LU preconditioner. However, the result ...
2
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1answer
113 views

LU Decomposition with memory-mapped matrices

I have a ~4.12 Tb structured relatively-sparse matrix dataset (about 8% of the matrix entries are non-zero) that i want to apply an LU decomposition, however, given the size of it, loading it in ...
6
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1answer
2k views

Implementing Explicit formulation of 1D wave equation in Matlab

So the theory is straightforward. We have: $$\frac{\partial^2U}{\partial t^2}=c^2 \frac{\partial^2U}{\partial x^2}$$ discretizing it gives: $$\frac{U(i+1,j)- 2U(i,j) + U(i-1,j)}{(\Delta t)^2} = c^2 ...
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661 views

Problems where Conjugate gradient works much better than GMRES

I am interested in cases where Conjugate gradient works much better than GMRES method. In general, CG is preferable choice in many cases of SPD (symmetric-positive-definite) because it requires less ...
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3answers
833 views

Solving a sparse and highly ill-conditioned system

I intend to solve Ax = b where A is complex, sparse, unsymmetric and highly ill-conditioned (condition number ~ 1E+20) square or rectangular matrix. I have been able to solve the system with ZGELSS in ...
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2answers
82 views

Problems where SPD linear system arises

I know some of the places where SPD linar systems arises such as elliptic PDEs and normal equations. Can I have a more comprehensive list of scientific applications which require solving SPD linear ...
3
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1answer
170 views

BFGS methods for constrained elasticity problems

My dear community, I am wondering why BFGS methods are not so widely used for simulating mechanical problems which heavily still relies on inverting the hessian matrix. I am essentially interested ...
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1answer
149 views

Are there any specialized methods available for solving structurally symmetric sparse linear systems?

When solving $Ax=b$, prior knowledge about $A$'s structure can help in designing an efficient solver which exploits this information (e.g conjugate gradient method is to be used when $A$ is ...