| Version: | 1.0-0 |
| Date: | 2025-07-17 |
| Title: | Eigendecomposition, Singular-Values and the Power Method |
| Author: | Doulaye Dembele 
[aut, cre] |
| Maintainer: | Doulaye Dembele <doulaye@igbmc.fr> |
| Depends: | R (≥ 4.0), |
| Description: | For a data matrix with m rows and n columns (m>=n), the power method is used to compute, simultaneously, the eigendecomposition of a square symmetric matrix. This result is used to obtain the singular value decomposition (SVD) and the principal component analysis (PCA) results. Compared to the classical SVD method, the first r singular values can be computed. |
| License: | GPL-2 | GPL-3 [expanded from: GPL (≥ 2)] |
| NeedsCompilation: | yes |
| Packaged: | 2025-07-17 08:32:06 UTC; doulaye |
| Repository: | CRAN |
| Date/Publication: | 2025-07-17 08:50:13 UTC |
Eigendecomposition, Singular-Values and the Power Method
Description
The power method is used to compute simultaneously the eigenvectors of a square symmetric matrix. Using the classical method, all eigenvectors are computed. The method used here allows to compute the first r eigenvectors using only matrix multiplications and the Gram-Schmidt orthogonalization algorithm. The relationships between the eigendecomposition factors, on the one hand, and the PCA factors or SVD factors, on the order hand, are used to get SVD or PCA results).
Details
| Package: | psvd |
| Type: | Package |
| Version: | 0.1-0 |
| Date: | 2024-10-02 |
| License: | GPL (>= 2) |
Package psvd has the following functions:
| calcSVD(): | Given a data matrix X of size (m,n), m >=n, this function allows to compute |
| the singular value decomposition. | |
| calcPCA(): | Given a data matrix X of size (m,n), m >=n, this function allows to\ compute |
| the principal component analysis. | |
| mGS(): | Modified Gramf-Schmidt orthogonalization method, R code, internal use. |
| mGSc(): | Modified Gramf-Schmidt orthogonalization method, C code, internal use. |
| eigenV(): | Computation of the eigenvectors matrix for a symmetric square matrix using |
| the power method, R Code, internal use. | |
| eigenVc(): | Computation of the eigenvectors matrix for a symmetric square matrix using |
| the power method, C Code, internal use. |
Author(s)
Doulaye Dembele: doulaye@igbmc.fr
References
Dembele D. (2024), Manuscript in preparation
Examples
data(iris)
X <- as.matrix(iris[,1:4])
rownames(X) <- iris[,5]
res <- calcSVD(X, r=4)
res$d
res$v
res$iter
Perform principal component analysis
Description
Given a data matrix, the function allows to perform principal component analysis using a power method to get the eigendecomposition.
Usage
calcPCA(X, r, eta, itmax, err, normed, mySeed)
Arguments
X |
Data matrix of size (m,n), m >= n. |
r |
Number of principal components, default: r=2. |
eta |
Power method tuning parameter, default: eta=10. |
itmax |
Maximum number of iteration in the power method, default: itmax=200. |
err |
Tolerance level in the power method, default: err=1e-8. |
normed |
TRUE (default) or FALSE for PCA using standardized data or not. |
mySeed |
An integer allowing to reproduce results from two different runs, default: mySeed=50. |
Details
X is usually a data matrix .
Value
This function returns a data frame containing 5 components
values |
Eigenvalues |
vectors |
Matrix with the eigenvectors. |
iter |
The number of iterations used in the eigendecomposition. |
li |
Projection of rows in the r principal components space. |
co |
Projection of columns in the r principal components space. |
Examples
data(iris)
X <- as.matrix(iris[,1:4])
rownames(X) <- iris[,5]
res <- calcPCA(X, r=3)
res$values
pcol <- c(rep("cyan",50), rep("red",50), rep("blue",50))
plot(res$li[,1], res$li[,3], col = pcol)
Perform singular values decomposition
Description
Given a data matrix, the function allows to perform a singular decomposition using a power method and relationship between SVD factors and the eigendecomposition factors.
Usage
calcSVD(X, r, eta, q, itmax, err, mySeed)
Arguments
X |
Data matrix of size (m,n), m >= n. |
r |
Rank r approximation, default: r=2. |
eta |
Power method tuning parameter, default: eta=10. |
q |
Power method tuning parameter, default: q=2. |
itmax |
Maximum number of iteration in the power method, default: itmax=200. |
err |
Tolerance level in the power method, default: err=1e-8. |
mySeed |
An integer allowing to reproduce results from two different runs, default: mySeed=50. |
Details
X is usually a data matrix.
Value
This function returns a data frame containing 4 components
d |
Singular values. |
u |
Matrix with the right eigenvectors. |
v |
Matrix with the right eigenvectors. |
iter |
The number of iterations used in the eigendecomposition. |
Examples
data(iris)
X <- as.matrix(iris[,1:4])
rownames(X) <- iris[,5]
res <- calcSVD(X, r=3)
res$d
res$v
res$iter
Compute the eigenvectors matrix of a square symmetric matrix
Description
This is an internal function which uses a R code to calculate an eidendecomposition of a square symmetric matrix. This function is used in the power method allowing to compute singular values and principal component analysis.
Usage
eigenV(xmat, wp, itmax, err)
Arguments
xmat |
Square symmetric matrix of order d. |
wp |
Columns orthogonal matrix of size (d,r), r <= d. |
itmax |
Maximum number of iterations. |
err |
Tolerance level in the iterative search. |
Value
This function returns a data frame containing 2 components
wc |
Eigenvectors matrix. |
iter |
Number of iterations by the power method. |
Examples
d <- 3
w <- matrix(rnorm(d*d,0,1), ncol=d)
wp <- mGS(w)
XtX <- matrix(c(3,2,1,2,1,0,1,0,1), ncol=3)
res <- eigenV(XtX, wp, itmax=100, err=1e-8)
t(res$wc)
Compute the eigenvectors of a square symmetric matrix
Description
This is an internal function which uses a C code to calculate an eidendecomposition of a square symmetric matrix. This function is used in the power method allowing to compute singular values and principal component analysis.
Usage
eigenVc(xmat, wp, d, r, itmax, err)
Arguments
xmat |
Square symmetric matrix of order d. |
wp |
Columns orthogonal matrix of size (d,r), r <= d. |
d |
Number of rows of wp. |
r |
Number of columns of wp. |
itmax |
Maximum number of iterations. |
err |
Tolerance level in the iterative search. |
Value
This function returns a data frame containing 2 components
wc |
Eigenvectors matrix. |
iter |
Number of iterations by the power method. |
Examples
d <- 3
r <- 3
w <- c(rnorm(d*r,0,1))
res <- mGSc(w, d, r)
wp <- res$wp
XtX <- c(3,2,1,2,1,0,1,0,1)
res <- eigenVc(XtX, wp, d, r, itmax=100, err=1e-8)
wc <- matrix(res$wc, d, r)
t(wc)
Modified Gram-Schmidt orthogonalization of a matrix
Description
This is an internal function which uses a R code to calculate an orthogonalization of a matrix. This function is used in the power method allowing to compute an eigendecomposition of a symmetric square.
Usage
mGS(A)
Arguments
A |
Matrix in vector form. |
Value
This function returns a matrix which columns are the eigenvectors
wp |
Eigevectors matrix. |
Examples
A <- matrix(rnorm(6,0,1), ncol=2)
res <- mGS(A)
t(res)
Modified Gram-Schmidt orthogonalization of a matrix
Description
This is an internal function which uses a C code to calculate an orthogonalization of a matrix. This function is used in the power method allowing to compute an eigendecomposition of a symmetric square.
Usage
mGSc(amat, m, n)
Arguments
amat |
Matrix in vector form. |
m |
Number of rows of the matrix amat. |
n |
Number of columns of the matrix amat. |
Value
This function returns a data frame containing 1 component
wp |
Eigenvectors matrix. |
Examples
d <- 3
r <- 2
amat <- c(rnorm(d*r,0,1))
res <- mGSc(amat, d, r)
wp <- matrix(res$wp, nrow=d, ncol=r)
t(wp)