Fast and Elegant Numerical Linear Algebra Using the RcppEigen Package

The RcppEigen package provides access from R (R Core Team 2023a) to the Eigen (Guennebaud, Jacob et al. 2012) C++ template library for numerical linear algebra. Rcpp (Eddelbuettel et al. 2024; Eddelbuettel 2013) classes and specializations of the C++ templated functions as and wrap from Rcpp provide the “glue” for passing objects from R to C++ and back. Several introductory examples are presented. This is followed by an in-depth discussion of various available approaches for solving least-squares problems, including rank-revealing methods, concluding with an empirical run-time comparison. Last but not least, sparse matrix methods are discussed.


Introduction
Linear algebra is an essential building block of statistical computing.Operations such as matrix decompositions, linear program solvers, and eigenvalue/eigenvector computations are used in many estimation and analysis routines.As such, libraries supporting linear algebra have long been provided by statistical programmers for different programming languages and environments.Because it is object-oriented, C++, one of the central modern languages for numerical and statistical computing, is particularly effective at representing matrices, vectors and decompositions, and numerous class libraries providing linear algebra routines have been written over the years.
As both the C++ language and standards have evolved (Meyers 2005(Meyers , 1995; ISO/IEC 2011), so have the compilers implementing the language.Relatively modern language constructs such as template meta-programming are particularly useful because they provide overloading of operations (allowing expressive code in the compiled language similar to what can be done in scripting languages) and can shift some of the computational burden from the run-time to the compile-time.(A more detailed discussion of template meta-programming is, however, beyond the scope of this paper).Veldhuizen (1998) provided an early and influential implementation of numerical linear algebra classes that already demonstrated key features of this approach.
Its usage was held back at the time by the limited availability of compilers implementing all the necessary features of the C++ language.This situation has greatly improved over the last decade, and many more libraries have been made available.One such C++ library is Eigen by Guennebaud, Jacob et al. (2012) which started as a sub-project to KDE (a popular Linux desktop environment), initially focussing on fixed-size matrices to represent projections in a visualization application.Eigen grew from there and has over the course of about a decade produced three major releases with Eigen3 being the current major version.To check the minor and patch version numbers, load the RcppEigen package and call this (internal) helper function: R> RcppEigen:::eigen_version(FALSE) major minor patch 3 3 5 Eigen is of interest as the R system for statistical computation and graphics (R Core Team 2023a) is itself easily extensible.This is particular true via the C language that most of R's compiled core parts are written in, but also for the C++ language which can interface with C-based systems rather easily.The manual "Writing R Extensions" (R Core Team 2023b) is the basic reference for extending R with either C or C++.
The Rcpp package by Eddelbuettel and François (2011); Eddelbuettel et al. (2024) facilitates extending R with C++ code by providing seamless object mapping between both languages.As stated in the Rcpp (Eddelbuettel et al. 2024) vignette, "Extending Rcpp" as well as in Eddelbuettel (2013) Rcpp facilitates data interchange between R and C++ through the templated functions Rcpp::as (for conversion of objects from R to C++) and Rcpp::wrap (for conversion from C++ to R).
The RcppEigen package provides the header files composing the Eigen C++ template library, along with implementations of Rcpp::as and Rcpp::wrap for the C++ classes defined in Eigen.
The Eigen classes themselves provide high-performance, versatile and comprehensive representations of dense and sparse matrices and vectors, as well as decompositions and other functions to be applied to these objects.The next section introduces some of these classes and shows how to interface to them from R.

R object type
Eigen class typedef numeric matrix MatrixXd integer matrix MatrixXi complex matrix MatrixXcd numeric vector VectorXd integer vector VectorXi complex vector VectorXcd Matrix::dgCMatrix SparseMatrix<double> Table 1: Correspondence between R matrix and vector types and classes in the Eigen namespace.
typedefs for common classes that correspond to R matrices and vectors, as shown in Table 1, and this paper will use these typedefs.
Here, Vector and Matrix describe the dimension of the object.The X signals that these are dynamically-sized objects (as opposed to fixed-size matrices such as 3 × 3 matrices also available in Eigen).Lastly, the trailing characters i, d and cd denote, respectively, storage types integer, double and complex double.
The C++ classes shown in Table 1 are in the Eigen namespace, which means that they must be written as Eigen::MatrixXd.However, if one prefaces the use of these class names with a declaration like using Eigen::MatrixXd; then one can use these names without the namespace qualifier.

Mapped matrices in Eigen
Storage for the contents of matrices from the classes shown in Table 1 is allocated and controlled by the class constructors and destructors.Creating an instance of such a class from an R object involves copying its contents.An alternative is to have the contents of the R matrix or vector mapped to the contents of the object from the Eigen class.For dense matrices one can use the Eigen templated class Map, and for sparse matrices one can deploy the Eigen templated class MappedSparseMatrix.
One must, of course, be careful not to modify the contents of the R object in the C++ code.
A recommended practice is always to declare mapped objects as const.

Arrays in Eigen
For matrix and vector classes Eigen overloads the '*' operator as matrix multiplication.
Occasionally component-wise operations instead of matrix operations are desired, for which the Array templated classes are used in Eigen.Switching back and forth between matrices or vectors and arrays is usually done via the array() method for matrix or vector objects and the matrix() method for arrays.

Structured matrices in Eigen
Eigen provides classes for matrices with special structure such as symmetric matrices, triangular matrices and banded matrices.For dense matrices, these special structures are described as "views", meaning that the full dense matrix is stored but only part of the matrix is used in operations.For a symmetric matrix one must specify whether the lower triangle or the upper triangle is to be used as the contents, with the other triangle defined by the implicit symmetry.

Some simple examples
C++ functions to perform simple operations on matrices or vectors can follow a pattern of: 1. Map the R objects passed as arguments into Eigen objects.
2. Create the result.
An idiom for the first step is using Eigen::Map; using Eigen::MatrixXd; using Rcpp::as; const Map<MatrixXd> A(as<Map<MatrixXd> >(AA)); where AA is the name of the R object (of type SEXP in C and C++) passed to the C++ function.
An alternative to the using declarations is to declare a typedef as in typedef Eigen::Map<Eigen::MatrixXd> MapMatd; const MapMatd A(Rcpp::as<MapMatd>(AA)); The cxxfunction function from the inline package (Sklyar, Murdoch, Smith, Eddelbuettel, François, Soetaert, and Ranke 2021) for R and its RcppEigen plugin provide a convenient method of developing and debugging the C++ code.For actual production code one generally incorporates the C++ source code files in a package and includes the line LinkingTo: Rcpp, RcppEigen in the package's DESCRIPTION file.The RcppEigen.package.skeletonfunction provides a quick way of generating the skeleton of a package that will use RcppEigen.
The cxxfunction with the "Rcpp" or "RcppEigen" plugins has the as and wrap functions already defined as Rcpp::as and Rcpp::wrap.In the examples below these declarations are omitted.It is important to remember that they are needed in actual C++ source code for a package.
The first few examples are simply for illustration as the operations shown could be more effectively performed directly in R. Finally, the results from Eigen are compared to those from the direct R results.

Transpose of an integer matrix
The next R code snippet creates a simple matrix of integers using Eigen::Map; using Eigen::MatrixXi; // Map the integer matrix AA from R const Map<MatrixXi> A(as<Map<MatrixXi> >(AA)); // evaluate and return the transpose of A const MatrixXi At(A.transpose()); return wrap(At);

R> (
and, in Figure 1, the transpose() method for the Eigen::MatrixXi class is used to return the transpose of the supplied matrix.The R matrix in the SEXP AA is first mapped to an Eigen::MatrixXi object, and then the matrix At is constructed from its transpose and returned to R.
The R snippet below compiles and links the C++ code segment.The actual work is done by the function cxxfunction from the inline package which compiles, links and loads code written in C++ and supplied as a character variable.This frees the user from having to know about compiler and linker details and options, which makes "exploratory programming" much easier.Here the program piece to be compiled is stored as a character variable named transCpp, and cxxfunction creates an executable function which is assigned to ftrans.This new function is used on the matrix A shown above, and one can check that it works as intended by comparing the output to an explicit transpose of the matrix argument.

Crossproduct of a single matrix
As shown in the last example, the R function crossprod calculates the product of the transpose of its first argument with its second argument.The single argument form, crossprod(X), evaluates X ⊤ X.One could, of course, calculate this product as t(X) %*% X but crossprod(X) is roughly twice as fast because the result is known to be symmetric and only one triangle needs to be calculated.The function tcrossprod evaluates crossprod(t(X)) without actually forming the transpose.
For triangular matrices the class is TriangularView and the method is triangularView.
For self-adjoint views the rankUpdate method adds a scalar multiple of AA ⊤ to the current symmetric matrix.The scalar multiple defaults to 1.The code in Figure 3 produces both A ⊤ A and AA ⊤ from a matrix A.

R> stopifnot(all.equal(crp[[1]], crossprod(A)), + all.equal(crp[[2]], tcrossprod(A)))
To some, the expressions in Figure 3 to construct AtA and AAt are compact and elegant.To others they are hopelessly confusing.If you find yourself in the latter group, you just need to read the expression left to right.So, for example, we construct AAt by creating a general integer matrix of size m × m (where A is m × n), ensuring that all its elements are zero, regarding it as a self-adjoint (i.e., symmetric) matrix using the elements in the lower triangle, then adding AA ⊤ to it and converting back to a general matrix form (i.e., the strict lower triangle is copied into the strict upper triangle).
In more detail: 1. MatrixXi(n, n) creates an n × n integer matrix with arbitrary contents  The assignment of this symmetric matrix to the (general) MatrixXi object AAt causes the result to be symmetrized during the assignment.
For these products one could define the symmetric matrix from either the lower triangle or the upper triangle as the result will be symmetrized before it is returned.
To cut down on repetition of using statements we gather them in a character variable, incl, that will be given as the includes argument in the calls to cxxfunction.We also define a utility function, AtA, that returns the crossproduct matrix as shown in Figure 4 3

.4. Cholesky decomposition of the crossprod
The Cholesky decomposition of the positive-definite, symmetric matrix, A, can be written in several forms.Numerical analysts define the "LLt" form as the lower triangular matrix, L, such that A = LL ⊤ and the "LDLt" form as a unit lower triangular matrix L and a diagonal matrix D with positive diagonal elements such that A = LDL ⊤ .Statisticians often write the decomposition as A = R ⊤ R where R is an upper triangular matrix.Of course, this R is simply the transpose of L from the "LLt" form.
The templated Eigen classes for the LLt and LDLt forms are called LLT and LDLT and the corresponding methods are .llt()and .ldlt().Because the Cholesky decomposition involves taking square roots, we pass a numeric matrix, A, not an integer matrix.

Determinant of the cross-product matrix
The "D-optimal" criterion for experimental design chooses the design that maximizes the determinant, |X ⊤ X|, for the n × p model matrix (or Jacobian matrix), X.The determinant, |L|, of the p × p lower Cholesky factor L, defined so that LL ⊤ = X ⊤ X, is the product of its diagonal elements, as is the case for any triangular matrix.By the properties of determinants, Alternatively, if using the "LDLt" decomposition, LDL ⊤ = X ⊤ X where L is unit lower triangular and D is diagonal then |X ⊤ X| is the product of the diagonal elements of D. Because it is known that the diagonal elements of D must be non-negative, one often evaluates the logarithm of the determinant as the sum of the logarithms of the diagonal elements of D. Several options are shown in Figure 6.

Least squares solutions
A common operation in statistical computing is calculating a least squares solution, β, defined as β = argmin where the model matrix, X, is n×p (n ≥ p) and y is an n-dimensional response vector.There are several ways, based on matrix decompositions, to determine such a solution.Earlier, two forms of the Cholesky decomposition were discussed: "LLt" and "LDLt", which can both be used to solve for β.Other decompositions that can be used are the QR decomposition, with or without column pivoting, the singular value decomposition and the eigendecomposition of a symmetric matrix.
Determining a least squares solution is relatively straightforward.However, statistical computing often requires additional information, such as the standard errors of the coefficient estimates.Calculating these involves evaluating the diagonal elements of X ⊤ X −1 and the residual sum of squares, ∥y − X β∥ 2 .

Least squares using the "LLt" Cholesky
Figure 7 shows a calculation of the least squares coefficient estimates (betahat) and the standard errors (se) through an "LLt" Cholesky decomposition of the crossproduct of the model matrix, X. Next, the results from this calculation are compared to those from the lm.fit function in R (lm.fit is the workhorse function called by lm once the model matrix and response have been evaluated).
R> lltLS <-cxxfunction(signature(XX = "matrix", yy = "numeric"), lltLSCpp, + "RcppEigen", incl) R> data("trees", package = "datasets") R> str lltLS(cbind(1,log(Girth)  There are several aspects of the C++ code in Figure 7 worth mentioning.The solve method for the LLT object evaluates, in this case, X X −1 X ⊤ y but without actually evaluating the inverse.The calculation of the residuals, y − y, can be written, as in R, as y -fitted.(But note that Eigen classes do not have a "recycling rule" as in R.That is, the two vector operands must have the same length.)The norm() method evaluates the square root of the sum of squares of the elements of a vector.Although one does not explicitly evaluate X ⊤ X −1 does evaluate L −1 to obtain the standard errors.Note also the use of the colwise() method in the evaluation of the standard errors.It applies a method to the columns of a matrix, returning a vector.The Eigen colwise() and rowwise() methods are similar in effect to the apply function in R. In the descriptions of other methods for solving least squares problems, much of the code parallels that shown in Figure 7.The redundant parts are omitted, and only the evaluation of the coefficients, the rank and the standard errors is shown.Actually, the standard errors are calculated only up to the scalar multiple of s, the residual standard error, in these code fragments.The calculation of the residuals and s and the scaling of the coefficient standard errors is the same for all methods.(See the files fastLm.hand fastLm.cpp in the RcppEigen source package for details.)

Least squares using the unpivoted QR decomposition
A QR decomposition has the form where Q is an n × n orthogonal matrix, which means that Q ⊤ Q = QQ ⊤ = I n , and the n × p matrix R is zero below the main diagonal.The n × p matrix Q 1 is the first p columns of Q and the p × p upper triangular matrix R 1 is the top p rows of R.There are three Eigen classes for the QR decomposition: HouseholderQR provides the basic QR decomposition using Householder transformations, ColPivHouseholderQR incorporates column pivots and FullPivHouseholderQR incorporates both row and column pivots.Figure 8 shows a least squares solution using the unpivoted QR decomposition.The calculations in Figure 8 are quite similar to those in Figure 7.In fact, if one had extracted the upper triangular factor (the matrixU() method) from the LLT object in Figure 7, the rest of the code would be nearly identical.

Handling the rank-deficient case
One important consideration when determining least squares solutions is whether rank(X) is p, a situation described by saying that X has "full column rank".When X does not have full column rank it is said to be "rank deficient".
Although the theoretical rank of a matrix is well-defined, its evaluation in practice is not.At best one can compute an effective rank according to some tolerance.Decompositions that allow to estimation of the rank of the matrix in this way are said to be "rank-revealing".
Because the model.matrixfunction in R does a considerable amount of symbolic analysis behind the scenes, one usually ends up with full-rank model matrices.The common cases of rank-deficiency, such as incorporating both a constant term and a full set of indicators columns for the levels of a factor, are eliminated.Other, more subtle, situations will not be detected at this stage, however.A simple example occurs when there is a "missing cell" in a two-way layout and the interaction of the two factors is included in the model.

R> dd <-data.frame(f1 = gl(4, 6, labels = LETTERS[1:4]), + f2 = gl(3, 2, labels = letters[1:3]))[-(7:8), ] R> xtabs(~f2 + f1, dd)
R> mm <-model.matrix(~f1* f2, dd) R> kappa(mm) [1] 4.30923e+16 This yields a large condition number, indicating rank deficiency.Alternatively, the reciprocal condition number can be evaluated.The lm function for fitting linear models in R uses a rank-revealing form of the QR decomposition.When the model matrix is determined to be rank deficient, according to the threshold used in R's qr function, the model matrix is reduced to rank (X) columns by pivoting selected columns (those that are apparently linearly dependent on columns to their left) to the right hand side of the matrix.A solution for this reduced model matrix is determined and the coefficients and standard errors for the redundant columns are flagged as missing.

R>
An alternative approach is to evaluate the "pseudo-inverse" of X from the singular value decomposition (SVD) of X or the eigendecomposition of X ⊤ X.The SVD is of the form where U is an orthogonal n × n matrix and U 1 is its leftmost p columns, D is n × p and zero off the main diagonal so that D 1 is a p × p diagonal matrix with non-increasing, non-negative diagonal elements, and V is a p × p orthogonal matrix.The pseudo-inverse of D 1 , written D + 1 is a p × p diagonal matrix whose first r = rank(X) diagonal elements are the inverses of the corresponding diagonal elements of D 1 and whose last p − r diagonal elements are zero.The tolerance for determining if an element of the diagonal of D 1 is considered to be (effectively) zero is a multiple of the largest singular value (i.e., the (1, 1) element of D).The pseudo-inverse, X + , of X is defined as In Figure 9 a utility function, Dplus, is defined to return the diagonal of the pseudo-inverse, D + 1 , as an array, given the singular values (the diagonal of D 1 ) as an array.Calculation of the maximum element of d (the method is called .maxCoeff()) and the use of a threshold() function provides greater generality for the function.It can be used on the eigenvalues of X ⊤ X, as shown in Section 4.5, even though these are returned in increasing order, as opposed to the decreasing order of the singular values.

Least squares using the SVD
With these definitions the code for least squares using the singular value decomposition can be written as in Figure 10.Even in the rank-deficient case this code will produce a complete set of coefficients and their standard errors.The user must be careful to check if the computed rank is less than p, the number of columns in X, in which case the estimated coefficients are just one of an infinite number of coefficient vectors that produce the same fitted values.It happens that the solution from this pseudo-inverse is the minimum norm solution (in the sense that ∥β∥ 2 is minimized among those β that minimize ∥y − Xβ∥ 2 ).
The standard errors of the coefficient estimates in the rank-deficient case must be interpreted carefully.The solution with one or more missing coefficients, as returned by the lm.fit function in R and by the column-pivoted QR decomposition described in Section 4.6, does not provide standard errors for the missing coefficients.That is, both the coefficient and its standard error are returned as NA because the least squares solution is performed on a reduced model matrix.It is also true that the solution returned by the SVD method is with respect to a reduced model matrix but the p coefficient estimates and their p standard errors don't show this.They are, in fact, linear combinations of a set of r coefficient estimates and their standard errors.

Least squares using the eigendecomposition
The eigendecomposition of X ⊤ X is defined as where V , the matrix of eigenvectors, is a p × p orthogonal matrix and Λ is a p × p diagonal matrix with non-decreasing, non-negative diagonal elements, called the eigenvalues of X ⊤ X.
When the eigenvalues are distinct, this V is the same (up to reordering of the columns) as that in the SVD and the eigenvalues of X ⊤ X are the (reordered) squares of the singular values of X.
With these definitions one can adapt much of the code from the SVD method for the eigendecomposition, as shown in Figure 11.

Least squares using the column-pivoted QR decomposition
The column-pivoted QR decomposition provides results similar to those from R in both the full-rank and the rank-deficient cases.The decomposition is of the form where, as before, Q is n × n and orthogonal and R is n × p and upper triangular.The p × p matrix P is a permutation matrix.That is, its columns are a permutation of the columns of I p .It serves to reorder the columns of X so that the diagonal elements of R are non-increasing in magnitude.
An instance of the class Eigen::ColPivHouseholderQR has a rank() method returning the computational rank of the matrix.When X is of full rank one can use essentially the same code as in the unpivoted decomposition except that one must reorder the standard errors.
When X is rank-deficient, the coefficients and standard errors are evaluated for the leading r columns of XP only.
In the rank-deficient case the straightforward calculation of the fitted values, as X β, cannot be used because some of the estimated coefficients, β, are NA so the product will be a vector of NA's.One could do some complicated rearrangement of the columns of X and the coefficient estimates but it is conceptually (and computationally) easier to employ the relationship The vector Q ⊤ y is called the "effects" vector in R.

Comparative speed
In the RcppEigen package the R function to fit linear models using the methods described above is called fastLm.It follows an earlier example in the Rcpp package which was carried over to both RcppArmadillo and RcppGSL.The natural question to ask is, "Is it indeed fast to use these methods based on Eigen?".To this end, the package provides benchmarking code for these methods, R's lm.fit function and the fastLm implementations in the RcppArmadillo (Eddelbuettel et al. 2023)  It can be run as R> source(system.file("examples","lmBenchmark.R", package = "RcppEigen")) Results will vary according to the speed of the processor and the implementation of the BLAS (Basic Linear Algebra Subroutines) used.(Eigen methods do not use the BLAS but the other methods do.)The Eigen3 template library does not use multi-threaded code for these operations but does use the graphics pipeline instructions (SSE and SSE2, in this case) in some calculations.
Results obtained on a desktop computer, circa 2010, are shown in modified to be rank-revealing.However, the dimensions of the problem will influence the comparative results.Because there are 100,000 rows in X, methods that decompose the whole X matrix (all the methods except those named above) will be at a disadvantage.
The pivoted QR method is 1.6 times faster than R's lm.fit on this test and provides nearly the same information as lm.fit.Methods based on the singular value decomposition (SVD and GSL) are much slower but, as mentioned above, this is caused in part by X having many more rows than columns.The GSL method from the GNU Scientific Library uses an older algorithm for the SVD and is clearly out of contention.
The GESDD method provides an interesting hybrid: It uses the Eigen classes, but then deploys the LAPACK routine dgesdd for the actual SVD calculation.The resulting time is much faster than using the SVD implementation of Eigen which is not a particularly fast SVD method.

Delayed evaluation
A form of delayed evaluation is used in Eigen.That is, many operators and methods do not evaluate the result but instead return an "expression object" that is evaluated when needed.
As an example, even though one writes the X ⊤ X evaluation as .rankUpdate(X.adjoint()) the X.adjoint() part is not evaluated immediately.The rankUpdate method detects that it has been passed a matrix that is to be used in its transposed form and evaluates the update by taking inner products of columns of X instead of rows of X ⊤ .
As Eigen has developed some of these unevaluated expressions have been modified.In Eigen 3.1, which is incorporated in version 0.3.1 of RcppEigen, the .adjoint()method applied to a real dense matrix copies the contents of the original matrix, flags it as row-major and interchanges the number of rows and columns.This is indeed the adjoint of the original matrix but, at one time, the wrap method for the Eigen dense matrix classes would fail on row-major matrices.
In the code for the transpose of an integer matrix shown in Figure 1 the transpose is assigned const MapMati A(as<MapMati>(AA)); return wrap(A.transpose()); to a MatrixXi object before applying wrap to it.The assignment forces the evaluation as a column-major matrix.In early versions of the RcppEigen package if this step is skipped, as in Figure 13, the result would have been incorrect.

Sparse matrices
Eigen provides sparse matrix classes.An R object of class dgCMatrix (from the Matrix package by Bates and Maechler (2023)) can be mapped as in Figure 14.

Summary
This paper introduced the RcppEigen package which provides high-level linear algebra computations as an extension to the R system.RcppEigen is based on the modern C++ library Eigen which combines extended functionality with excellent performance, and utilizes Rcpp to interface R with C++.Several illustrations covered common matrix operations and several approaches to solving a least squares problem-including an extended discussion of rankrevealing approaches.Sparse matrix computations were also discussed, and a short example provided an empirical demonstration of the excellent run-time performance of the RcppEigen package.

Figure 2 :
Figure 2: prodCpp: Product and cross-product of two matrices.

Figure 3 :
Figure 3: crossprodCpp: Cross-product and transposed cross-product of a single matrix.

Figure 4 :
Figure4: The contents of the character vector, incl, that will preface C++ code segments that follow.

inlineFigure 9 :
Figure 9: DplusCpp: Create the diagonal d + of the pseudo-inverse, D + 1 , from the array of singular values, d.

Figure 13 :
Figure 13: badtransCpp: Transpose producing a run-time error in early versions of RcppEigen.
does indeed provide the desired answer The rank-revealing SVD method produces the same fitted values but not the same coefficients.
and RcppGSL (Eddelbuettel and François 2021) packages, if they are installed.The benchmark code, which uses the rbenchmark (Kusnierczyk 2012) package, is in a file named lmBenchmark.R in the examples subdirectory of the installed RcppEigen package.

Table 2 .
The processor used for these timings is a 4-core processor but almost all the methods are single-threaded and not affected by the number of cores.Only the arma, lm.fit, GESDD and GSL methods benefit from the multi-threaded BLAS implementation provided by OpenBLAS, and the relative speed increase is modest for this problem size and number of cores (at 7.76 seconds relative to 10.29 seconds for arma, 13.74 seconds relative to 16.91 seconds for lm.fit, and 176.95 seconds relative to 193.29 seconds for the GSL).Parallel computing approaches will always have to trade-off increased communication and overhead costs against the potential gains from running multiple execution threads.
These results indicate that methods based on forming and decomposing X ⊤ X (LDLt, LLt and SymmEig) are considerably faster than the others.The SymmEig method, using a rankrevealing decomposition, would be preferred, although the LDLt method could probably be

Table 2 :
lmBenchmark results on a desktop computer for the default size, 100, 000 × 40, fullrank model matrix running 20 repetitions for each method.Times (Elapsed, User, and Sys) are in seconds.The BLAS in use is a locally-rebuilt version of the OpenBLAS library included with Ubuntu 11.10.