Scientific programming hindawi publishing corporation. New parallel rankrevealing qr factorization algorithms. That allows us to look at the diagonal elements to figure out when to cut things off. The most accurate rank revealing factorization is, of course, the singular value. Of course there is much good matlab software but it does not satisfy the criteria that we have chosen. We call the factorization 1 a rankrevealing qr rrqr factorizationif it satisfies. On the failure of rank revealing qr factorization software a case study 2006 cached. The ones marked may be different from the article in the profile. In this paper we introduce carrqr, a communication avoiding rank revealing qr.
On the failure of rankrevealing qr factorization softwarea case study. Qr decomposition is often used to solve the linear least squares problem and is the basis for a particular eigenvalue algorithm, the qr algorithm. The 30year old problem, dating back to linpack, has undetectedly badly affected many computational routines and software packages, as well as the study of rankrevealing qr factorizations. On the failure of rankrevealing qr factorization software a case study article in acm transactions on mathematical software 352 july 2008 with 30 reads how we measure reads. Within each frontal matrix, lapack and the multithreaded blas enable the method to obtain high performance on. In linear algebra, a qr decomposition, also known as a qr factorization or qu factorization is a decomposition of a matrix a into a product a qr of an orthogonal matrix q and an upper triangular matrix r. The matrix q is orthogonal and r is upper triangular, so a qr is the required qrdecomposition. International journal for computational methods in engineering science and mechanics 16.
In section 2, we give an overview of the previous results on rank revealing lu and qr factorizations. A very useful factorization of an m by n m n matrix a is the qr factorization, given by aii qr, where ii e rnxn is a permutation matrix, qe rmx has orthogonal columns and satisfies qq i,, and r e rnx is upper triangular. It can also be used to estimate rank of matrix rank revealing qr decomposition. Rankone update to a rank revealing qr rrqr factorization. Lu factorization with panel rank revealing pivoting and. Mathworks is the leading developer of mathematical computing software for. Finally, x6 recalls the importance of pivoting from the numerical point of view. Rank revealing factorizations and low rank approximations.
Lu factorization with panel rank revealing pivoting and its. On the failure of rankrevealing qr factorization software. In this paper we introduce carrqr, a communication avoiding rank revealing qr factorization with tournament pivoting. This paper and the accompanying algorithm describe and analyze a suite of codes that implement combinations and modifications of several previously published methods for rrqr. Their combined citations are counted only for the first article. We present a novel method for the qr factorization of large tallandskinny matrices that introduces an approximation technique for computing the householder vectors. Its utility arises from the numerical stability offered by orthogonal transformations. Pdf we present a block algorithm for computing rankrevealing qr. A rankrevealing qr rrqr factorization is an efficient way to compute a reasonable representation of the null space of a matrix. We develop algorithms and implementations for computing rankrevealing qr rrqr factorizations of dense matrices. Rankrevealing qr factorization file exchange matlab. Test matrices 1 through 5 were designed to exercise column pivoting.
Implementation of jacobi rotations for accurate singular value computation in floating point arithmetic, siam j. Drmac z and bujanovic z 2008 on the failure of rankrevealing qr factorization software a case study, acm transactions on mathematical software toms, 35. Pdf a block algorithm for computing rankrevealing qr. Siam journal on numerical analysis volume, issue 6 10.
The singular value decomposition can be used to generate an rrqr, but it is not an efficient method to do so. Multifrontal multithreaded rankrevealing sparse qr factorization. Lowrank matrices arise in many scientific and engineering computations. Qr decomposition is often used to solve the linear least squares problem and is the basis for a particular eigenvalue algorithm, the qr. Strong rank revealing lu factorizations sciencedirect. On the failure of rank revealing qr factorization software. Plan low rank matrix approximation rank revealing qr factorization lu crtp. Key words, orthogonal factorization, rank revealing factorization, numericalrank. On the failure of rank revealing qr factorization software the netlib.
The problem of finding a rank revealing qr rrqr factorisation of a matrix a consists of permuting the columns of a such that the resulting qr factorisation contains an upper triangular matrix who. Both computational and storage costs of manipulating such matrices may be reduced by taking advantages of their lowrank properties. This also calls for serious rethinking of the compiler and debugger design for. San jose state university sjsu scholarworks masters theses masters theses and graduate research 1997 rank revealing qr factorizations lily l.
Communication avoiding rank revealing qr factorization. The routine uses a qr factorization with column pivoting. We show that carrqr reveals the numerical rank of a matrix in an analogous way to qr factorization with column pivoting qrcp. On the failure of rank revealing qr factorization software a case study. This paper reports an unexpected and rather erratic behavior of the lapack software implementation of the qr factorization with busingergolub column pivoting. On the failure of rank revealing qr factorization software citeseerx. Qr decomposition for rankdeficient matrix mathematics. Numerical algorithms for highperformance computational. On the failure of rankrevealing qr factorization software a case study, acm trans. Factorizations that are usually rank revealing but have a possibility of failure.
Rank revealing factorizations based on qr factorization with column permutations or lu factorization with row and column permutations are also efficient approaches for computing a lowrank approximation. Pdf multifrontal multithreaded rankrevealing sparse qr. Van straalen 20 experiences from software engineering of large scale amr multiphysics code frameworks, at the first workshop on sustainable software for science. Jan 24, 2020 we present a novel method for the qr factorization of large tallandskinny matrices that introduces an approximation technique for computing the householder vectors. Let m qr be the qr factorization of m with its columns permuted according to the permutation matrix. Wilkinson submitted by jack dongarra abstract an algorithm is presented for computing a column permutation li and a qr factorization a ii qr of an m by n m n matrix a such that a. The permutation returned by the panel rank revealing factorization is applied on the rows of the input matrix, and the lfactor of the panel is computed based on the rfactor of the strong rrqr factorization. While this greedy algorithm is known to fail on the socalled kahan. Suitesparseqr is a sparse qr factorization package based on the multifrontal method. Qr factorization, rank revealing, column pivoting, minimize communication. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Dec 09, 2019 on the failure of rankrevealing qr factorization software a case study, acm trans. We develop algorithms and implementations for computing rank revealing qr rrqr factorizations of dense matrices.
To compute a lowrank approximation of a dense matrix, in this paper, we study the performance of qr factorization with column pivoting or with restricted pivoting on. The need for single vector pivoting in classical algorithms for computing qr factorizations is avoided by the use of randomized sampling to find blocks of pivot vectors at once. Improve the ability of software developers to adapt existing software to new hardware. To compute a lowrank approximation of a dense matrix, in this paper, we study the performance of qr factorization with column pivoting or with restricted pivoting on multicore cpus with a. Demmel, laura grigoriy, ming gu z, and hua xiang x abstract. This approach is very competitive on a hybrid platform equipped with a graphics processor, with a performance advantage over the conventional factorization due to the reduced amount of data transfers between the graphics. An rrqr factorization or rank revealing qr factorization is a matrix decomposition algorithm based on the qr factorization which can be used to determine the rank of a matrix. Strong rankrevealing qr factorization math berkeley. Mechanical engineering, university of california, santa barbara verified email at engr. In this paper, we show how to construct a qr factorization which can also reveal the rank reliably. Rankrevealing qr factorization file exchange matlab central. This cited by count includes citations to the following articles in scholar. Practice and experiences wssspe1, november 17, denver, co.
Computing rankrevealing qr factorizations of dense matrices. If we take a qr factorization with pivoting, the diagonal elements of r are in decreasing order. For the qr factorization using classic gramschmidt algorithm, i found the 2 different implementations below. Existence of generalized mt factorizations in this section, we first define the generalized lur factorization of a given matrix a e r, and then we give an equivalence condition for the existence of the lur factorization. The mexfunctions are using the rankrevealing qr routines xgeqpx and xgeqpy from acm algorithm 782. Mathematics and computer science division, argonne national laboratory, 9700 s. Key words, orthogonal factorization, rankrevealing factorization, numericalrank. The algorithm uses a reduction strategy to perform the factorization which in turn allows a good degree of parallelism. Chan has noted that, even when the singular value decomposition of a matrix a is known, it is still not obvious how to find a rankrevealing qr factorization rrqr of a if a has numerical rank deficiency.
First, we develop an efficient block algorithm for approximating an rrqr factorization, employing a windowed version of the commonly used golub pivoting strategy, aided by incremental condition estimation. The use of householder transformations is inherently the most simple of the numerically stable qr decomposition algorithms due to the use of reflections as the mechanism for producing zeroes in the r matrix. Communication avoiding rank revealing qr factorization with. Although the upper bound of a quantity involved in the characterization of a rank revealing. A pivoting strategy based on the idea of local maximum volumes is introduced and a proof of the. To compute optionally a rank revealing qr factorization of a real general mbyn real or complex matrix a, which may be rank deficient, and estimate its effective rank using incremental condition estimation. This examples should convince the reader that the problem is serious, and that it may lead to numerical catastrophes in engineering applications. The problem of finding a rankrevealing qr rrqr factorisation of a matrix a. An rrqr factorization or rankrevealing qr factorization is a matrix decomposition algorithm based on the qr factorization which can be used to determine the rank of a matrix. The usual tradeoff is that the svd is more expensive but can reveal rank more reliably.
Software practicalities are emphasized throughout, with particular reference to lapack and matlab. Zlatko drmac and zvonimir bujanovic, title on the failure of rank revealing qr factorization software a case study, year 2006 share. Accuracy and stability of numerical algorithms guide books. Tallandskinny qr factorization with approximate householder. The roster of standard matrix decompositions includes the pivoted qr factorization, the eigenvalue decomposition. This fixes a subtle numerical bug caused by underestimate of the distance to singularity dating back to linpack that can give completely wrong results. Computer science division and mathematics department, uc. On the failure of rankrevealing qr factorization software a case study. In the rank one deficient case, the two algorithms produce the. Bischof, mathematics and computer science division, argonne.
Efficient algorithms for computing a strong rankrevealing qr. May 03, 20 in this paper we introduce carrqr, a communication avoiding rank revealing qr factorization with tournament pivoting. The first factorization of this type, the qr factorization with column pivoting, referred to as qrcp, was introduced in. The svd can be used to generate an rrqr, but it is not an efficient method to do so. Can also handle complex and single precision arrays. Rank revealing factorizations, and low rank approximations l. Both the singular value decomposition svd and the qr factorization play central roles in signal processing algorithms. Efficient algorithms for computing a strong rankrevealing. We show that carrqr reveals the numerical rank of a.
Wilkinson submitted by jack dongarra abstract an algorithm is presented for computing a column permutation li and a qr factorization a ii qr of an m by n m n matrix a such that a possible rank deficiency of a will be revealed. On the failure of rank revealing qr factorization software a case study zlatko drmac and zvonimir bujanovi c department of mathematics, university of zagreb, 0 zagreb, croatia. This approach is very competitive on a hybrid platform equipped with a graphics processor, with a performance advantage over the conventional factorization due to the reduced amount of data. Computing rankrevealing qr factorizations of dense.
Truncated lu factorization with column and row tournament pivoting experimental results, lu crtp. To compute optionally a rankrevealing qr factorization of a real general mbyn real or complex matrix a, which may be rankdeficient, and estimate its effective rank using incremental condition estimation the routine uses a qr factorization with column pivoting. Mathematical software new more robust and reliable xgeqp3 and xgeqpf routines qr factorization with column pivoting in lapack 3. Rank revealing qr factorizations sjsu scholarworks.
Efficient algorithms for computing a strong rankrevealing qr factorization. Drmac z and bujanovic z 2008 on the failure of rankrevealing qr factorization software a case. The algorithm is blocked, to allow it to be implemented efficiently. The first one uses the for loop to compute the upper. Matrix 6 was designed to test the behavior of the condition estimation in the presence of clusters for the smallest singular value. Theres probably a better way to do this with a rank revealing qr, but the following will work in a pinch. On the failure of rankrevealing qr factorization softwarea case study z drmac, z bujanovic acm transactions on mathematical software toms 35 2, 128, 2008. Qr factorization, rankrevealing orthogonal factorization. We combine computer experiments and numerical analysis to isolate, analyze, and fix the problem. Qr decomposition of a large sparse matrix is used to solve the linear least squares ls problem arising in power system state estimation. By applying another in essence also rankrevealing orthogonal factorization.