How To Properly Use ARPACK's Dsaupd And Dseupd?
Introduction
ARPACK (ARnoldi PACKage) is a software library for solving large-scale eigenvalue problems. It is widely used in various fields, including linear algebra, numerical analysis, and scientific computing. In this article, we will focus on using ARPACK's dsaupd and dseupd functions to solve symmetric eigenvalue problems. We will also provide a Rust example to demonstrate how to use these functions.
What are dsaupd and dseupd?
dsaupd and dseupd are two functions in ARPACK that are used to solve symmetric eigenvalue problems. dsaupd is a driver routine that uses the Arnoldi iteration to compute a few eigenvalues and corresponding eigenvectors of a symmetric matrix. dseupd is a utility routine that is used to extract the eigenvalues and eigenvectors from the result of dsaupd.
How to Use dsaupd and dseupd
To use dsaupd and dseupd, you need to follow these steps:
- Initialize the problem: You need to initialize the problem by calling the dsaupd function with the following parameters:
ID: The ID of the problem. It can be one of the following values:Isym: Symmetric eigenvalue problem.Asym: Asymmetric eigenvalue problem.
N: The size of the matrix.which: The type of eigenvalues to compute. It can be one of the following values:LM: Compute the smallest eigenvalues.SM: Compute the largest eigenvalues.LM: Compute the smallest eigenvalues.LM: Compute the largest eigenvalues.
nev: The number of eigenvalues to compute.tol: The tolerance for the convergence.resid: The residual norm.v: The initial guess for the eigenvectors.
- Compute the eigenvalues and eigenvectors: Call the dsaupd function with the following parameters:
ID: The ID of the problem.N: The size of the matrix.which: The type of eigenvalues to compute.nev: The number of eigenvalues to compute.tol: The tolerance for the convergence.resid: The residual norm.v: The initial guess for the eigenvectors.iparam: The parameters for the Arnoldi iteration.ipntr: The pointers to the workspace.workd: The workspace.rwork: The workspace.
- Extract the eigenvalues and eigenvectors: Call the dseupd function with the following parameters:
ID: The ID of the problem.N: The size of the matrix.which: The type of eigenvalues to compute.nev: The number of eigenvalues to compute.tol: The tolerance for the convergence.resid: The residual norm.v: The initial guess for the eigenvectors.iparam: The parameters for the Arnoldi iteration.ipntr: pointers to the workspace.workd: The workspace.rwork: The workspace.
Rust Example
Here is a Rust example that demonstrates how to use dsaupd and dseupd to solve a symmetric eigenvalue problem:
fn faer_arpack_symmetric_f64<F>(
mut transform: F,
dimension: usize,
which: &str,
nev: usize,
tol: f64,
resid: f64,
v: Vec<f64>,
iparam: Vec<i32>,
ipntr: Vec<i32>,
workd: Vec<f64>,
rwork: Vec<f64>,
) -> (Vec<f64>, Vec<f64>)
where
F: FnMut(&mut [f64]) -> Vec<f64>,
{
// Initialize the problem
let id = 1; // Isym
let n = dimension;
let which = which;
let nev = nev;
let tol = tol;
let resid = resid;
let v = v;
// Compute the eigenvalues and eigenvectors
let mut iparam = vec![0; 11];
let mut ipntr = vec![0; 14];
let mut workd = vec![0.0; 7 * n];
let mut rwork = vec![0.0; 3 * n];
dsaupd(
&mut id,
&mut n,
which,
&mut nev,
&mut tol,
&mut resid,
&mut v,
&mut iparam,
&mut ipntr,
&mut workd,
&mut rwork,
);
// Extract the eigenvalues and eigenvectors
let mut eigenvalues = vec![0.0; nev];
let mut eigenvectors = vec![vec![0.0; n]; nev];
dseupd(
&mut id,
&mut n,
which,
&mut nev,
&mut tol,
&mut resid,
&mut v,
&mut iparam,
&mut ipntr,
&mut workd,
&mut rwork,
&mut eigenvalues,
&mut eigenvectors,
);
(eigenvalues, eigenvectors)
}
Conclusion
In this article, we have discussed how to properly use ARPACK's dsaupd and dseupd functions to solve symmetric eigenvalue problems. We have also provided a Rust example that demonstrates how to use these functions. By following the steps outlined in this article, you can use dsaupd and dseupd to solve a wide range of eigenvalue problems.
References
Future Work
In the future, we plan to extend the Rust ARPACK bindings to support more features and functions. We also plan to provide more examples and tutorials to help users get started with using ARPACK in Rust.
Acknowledgments
Introduction
In our previous article, we discussed how to properly use ARPACK's dsaupd and dseupd functions to solve symmetric eigenvalue problems. In this article, we will answer some frequently asked questions about these functions.
Q: What is the difference between dsaupd and dseupd?
A: dsaupd is a driver routine that uses the Arnoldi iteration to compute a few eigenvalues and corresponding eigenvectors of a symmetric matrix. dseupd is a utility routine that is used to extract the eigenvalues and eigenvectors from the result of dsaupd.
Q: How do I choose the value of nev?
A: The value of nev is the number of eigenvalues to compute. It should be a positive integer less than or equal to the size of the matrix. A good choice for nev is to set it to a small number, such as 5 or 10, and then increase it as needed.
Q: What is the purpose of the resid parameter?
A: The resid parameter is the residual norm. It is used to determine when the Arnoldi iteration has converged. A small value of resid indicates that the iteration has converged.
Q: How do I choose the value of tol?
A: The value of tol is the tolerance for the convergence. It should be a small positive number. A good choice for tol is to set it to a value such as 1e-6 or 1e-8.
Q: What is the purpose of the v parameter?
A: The v parameter is the initial guess for the eigenvectors. It should be a vector of length n, where n is the size of the matrix. A good choice for v is to set it to a random vector.
Q: How do I use the iparam and ipntr parameters?
A: The iparam and ipntr parameters are used to control the Arnoldi iteration. iparam is a vector of length 11, where each element represents a different parameter of the iteration. ipntr is a vector of length 14, where each element represents a different pointer to the workspace. A good choice for iparam and ipntr is to set them to their default values.
Q: What is the purpose of the workd and rwork parameters?
A: The workd and rwork parameters are used to store the workspace for the Arnoldi iteration. workd is a vector of length 7n, where n is the size of the matrix. rwork is a vector of length 3n. A good choice for workd and rwork is to set them to their default values.
Q: How do I extract the eigenvalues and eigenvectors from the result of dsaupd?
A: To extract the eigenvalues and eigenvectors from the result of dsaupd, you can use the dseupd function. This function takes the result of dsaupd as input and returns the eigenvalues and eigenvectors.
Q: What are some common errors that can occur when using dsaupd and dseupd?
A: Some common errors that can occur when using dsaupd and dseupd include:
- The Arnoldi iteration has not converged.
- The eigenvalues and eigenvectors have not been computed correctly.
- The workspace has not been initialized correctly.
Conclusion
In this article, we have answered some frequently asked questions about ARPACK's dsaupd and dseupd functions. We hope that this article has been helpful in answering your questions and providing a better understanding of how to use these functions.
References
Future Work
In the future, we plan to extend the Rust ARPACK bindings to support more features and functions. We also plan to provide more examples and tutorials to help users get started with using ARPACK in Rust.
Acknowledgments
We would like to thank the ARPACK developers for creating such a powerful and widely used software library. We would also like to thank the Rust community for a great platform for building and sharing software.