Then the column space of C(M) is. vectors. In a nutshell, decomposition helps us to write optimistic algorithms, and calculate Linear Algebra Object properties and various matrices. Cholesky decomposition Conceptually the simplest computational method of spectral factorization might be ``Cholesky decomposition.'' For example, the matrix of ( 13 ) could have been found by Cholesky factorization of ( 12 ). Notation Row rank of M dim (R(M)) and Column Rank of M is dim(C(M)). As a result, and are vectors of length n-1 , and and are . // MPI_Bcast(Ajk.data(), Ajk.size(), MPI_DOUBLE, owner. but the solution would diverge. The right-looking algorithm for implementing this operation can be described by partitioning the matrices where and are scalars. For the multivariate normal distribution, the variance is a matrix (unfortunately . In summary, LU Decomposition can be done as, Major applications of LU decomposition are. Rank-factorization of a matrix is not unique. Cholesky decomposition is approximately 2x faster than LU Decomposition, where it applies. Suppose B is an identity matrix I, then the problem reduces to traditional eigenvalue problem. Eigenvectors are the factors. The Cholesky decomposition or Cholesky factorization is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose. unblocked algorithm or by calling the blocked Cholesky factorization algorithm recursively. ``Top-Left'', ``Bottom-Left'', and ``Bottom-Right'', respectively. Notice that if exists within one node, then two-dimensional mesh is linearized by ordering the nodes in takes advantage of this fact. to recursively build the elements of from the elements of .Likewise for a matrix, etc. I implemented some sequential matrix algorithms which act as basic matrix operations. all computation required to factor Cholesky Matrix Cholesky algorithm Cholesky factorization Cholesky matrix indefinite matrix negative definite matrix negative semidefinte matrix. Below is the description of the algorithm I wrote: In the left-looking algorithm, the processor has $row(j)$ broadcast all $A_{jj}$s left matrix and bro broadcast $A_{jj}^T$when $A_{jj}$ is updated. Below is the execution time of the algorithms: The block size for weak scalability is still 32 x 32 and run on a single node. Note that, when A is positive definite the Cholesky factor is given by. The Cholesky decomposition (or the Cholesky factorization) is a decomposition of a symmetric positive definite matrix into the product , where the factor is a lower triangular matrix with strictly positive diagonal elements. It was discovered by Andr-Louis Cholesky . For right-looking, the task is parallels in the inner-most loop. (solved as triangular solve with multiple-right-hand-sides does not produce a useful result if we stop part way to completion, Return the Cholesky decomposition, L * L.H, of the square matrix a, where L is lower-triangular and .H is the conjugate transpose operator (which is the ordinary transpose if a is real-valued).a must be Hermitian (symmetric if real-valued) and positive-definite. The positive-definiteness, is what ensures that a[k,k] is a positive number and sqrt is ok (see, for example, a Wiki explanation on that).. A similar story is happening with an incomplete Cholesky factorization, as its applicability is also limited to . then the first backsubstitution is filtering down a helix ! Every symmetric, positive definite matrix A can be decomposed into a product of a unique lower triangular matrix L and its transpose: <math>A = LL^T</math> The matrix inverse is required in the blocked Cholesky decomposition and only needed by the diagonal sub-matrixes that are lower triangular matrix. Positive definite matrices can be expressed in the form for a non-singular matrix X. In order to solve for the lower triangular matrix, we will make use of the Cholesky-Banachiewicz Algorithm. real*8 vectors length n. ! 3 0 obj << After data distributed, the full matrix is released to ensure enough memory for rank 0. Representing any matrix A as a product of three matrices: How SVD Algorithm factorize the Original Matrix: Schur Decomposition is based on Eigen Decomposition and is decomposed into 3 matrices of the Original matrix. Let us define few concepts to learn Rank Factorization. In this method, vectors to be considered as column of the matrix M. That is, , it is easy to write the QR Factorization, Consider the Matrix M of 3 rows and 3 columns. PLAPACK. The recursive algorithm starts with i := 1 and To do so, Decomposition methods are used to calculate determinant, upper and lower triangle matrices, matrix inversion, eigen values and eigen vectors, etc., to work on various types of matrices (symmetric, non-symmetric, square, non-square ). QR decomposition of a Matrix A into a product A = QR of an orthogonal matrix Q and an upper triangular matrix R. It is often used to solve linear least squares problem and is the basis for a particular eigenvalue algorithm, the QR algorithm. Furthermore, the multivector distribution is a The right-looking algorithm for implementing this operation can be described Data Scientists must think like an artist when finding a solution when creating a piece of code. Whenever applicable, the Cholesky algorithm is . Actually, each of those routines M~ 2qs9SzS>7Gz1O! U, which is exactly the same. The Cholesky factor exists i Ais positive de nite; in fact, the usual way to test numeri-cally for positive de niteness is to attempt a Cholesky factorization and see whether the algorithm succeeds or fails. So round j for these processors are % Algorithm 2.7 Heath, p.86. The block size is 32 x 32, which takes 8KB to fit in the L1 cache of one core. It suffices to know that object A references the original then accomplished by partitioning the vector in blocks Unlike weak scalability, I carefully choose the matrix size so that the total operations in the algorithm scales linearly with the number of the cores. Data Architect, Researcher in AI & Quantum Computing. The reason is due to the cost of communication. Note that there is a hidden $O(\log N_p)$ time for broadcast propagation upon the first receive of $A_{jk}$. Consider Matrix M and it has the form MX=b; By def M = LU and substitute in MX=b; then we Get LUX = b; Decompose Matrix M into Lower and upper triangle matrices as follows. This decomposition is known as the Cholesky decompostion, and L may be interpreted as the 'square root' of the matrix A. Let's assume we have a correlation matrix of 4 underlying assets: Using Cholesky decomposition, the lower triangular matrix is: The first column calculation: 1.00=sqrt(1) 0.80=0.8/1.00 0.20=0.2/1.00 0.50=0.5/1.00 Cholesky decomposition or factorization is a form of triangular decomposition that can only be applied to either a positive definite symmetric matrix or a positive definite Hermitian matrix. In the following equations explained eigenvectors ,eigenvalues and properties of decomposition. Matrix elements are not always numbers. 2(a)-(b) illustrate how the developed The Speedup is calculated with the sequential baseline by setting the number of processes to 1 in mpiexec. What is the other name for Cholesky factorization? take the minimum of. %PDF-1.4 Cholesky Decomposition. The data structure to store a matrix $A_{ij}$ is using STL std::vector instead of c array. The beauty of the Cholesky method is that it is numerically stable and accurate (as noted by Turing 1948) while requiring fewer floating-point operations and less workspace (computer memory) than alternative methods. Figure 4 (a) shows the Cholesky . using a one-dimensional data distribution. The best exact algorithm known for computing the determinant of general matrices, the Cholesky factorization, runs in a cubic complexity O(n 3). As with C array, the memory layout of C++ STL vector is linear and aligned, so there wont be any performance penalty. Still, inner-most is dominated by the processor having column j. These equations can be solved to yield R a column at a time, according to the following algorithm: The above algorithm requires flops and n square roots, where a flop is any of the four elementary scalar arithmetic operations +,-,*, and /. Let M be a matrix with dimensions (m,n). Global full matrix generation is done on rank 0 at first, then distributed by raw/column cyclic way. Then the rank of M is is at most k. Linear Algebra is the most essential subject in Algorithms, Computation (Classical and Quantum Computers) and storage space while computing. It was discovered by Andr-Louis Cholesky for real matrices. for the vector of the Cholesky method. A symmetric matrix A is said to be positive definite if for any non-zero x. and are matrices of size .The indicates the symmetric part of A , which will not Cholesky decomposition is an efficient method for inversion of symmetric positive-definite matrices. ; Calculating off-diagonal elements g i,j i > j (steps 2, 3 and 5) entails dividing some number by the last-calculated diagonal element. That routine itself resembles the presented And each round is dominated by the processor which owns the column j. Artists enjoy working on interesting problems, even if there is no obvious answer linktr.ee/mlearning Follow to join our 28K+ Unique DAILY Readers . One of the most widely used decomposition method is Eigen decomposition, decomposing a matrix into a set of eigenvectors and eigenvalues. of a Level-2 (left) and Level-3 (right) BLAS based right-looking Cholesky factorization. basic algorithm given above, Thanks for reading my article. Its retrieved by reference so there is no copy. (We looked at a special case of backsubstitution Principal Component Analysis is the tool in Exploratory Analysis , Dimensionality Reduction and Prediction models. by partitioning the matrices Cholesky decomposition From Rosetta Code Cholesky decomposition You are encouraged to solve this task according to the task description, using any language you may know. to an logical mesh of nodes using Step2. Decomposition can be computed by a form of Gaussian elimination that takes advantage of the Symmetry and definiteness. If D is allowed to have non-positive diagonal entries, the factorization exists for some indefinite matrices. The algorithm terminates once the pivot is less than tol. In the similar way M has left inverse. Cholesky decomposition is implemented in the Wolfram Language as CholeskyDecomposition [ m ]. A_{32} = A_{32}A_{22}^{-T}\). I use C++ to implement the parallel algorithm. For example: In modern libraries such as LAPACK,3 the factorization is implemented in partitioned form, which introduces another level of looping in order to extract the best performance from the memory hierarchies of modern computers. mesh of nodes[5,11,13,15]. We can see that the right-looking algorithm requires matrix multiplication between $A_{jj}$to all other sub-matrixes in the same column. Kindly note that, there are other decompositions exist in Linear Algebra and not covered in this article. Now, Being a matrix method, the Cholesky method of factorization The outcome matrices are the factors. Create a vector for the right-hand side of the equation Ax = b. b = sum (A,2); Since A = R T R with the Cholesky decomposition, the linear equation becomes R T R x = b. We cannot simply solve Poisson's equation as if is duplicated within the column that owns The code can be found on github: https://github.com/jaxonwang/parallel_numeric_class/tree/master/a3. MHB RSA algorithm to encrypt "abcdefghij" Last Post; Nov 29, 2021; Replies 0 Views 280. It decompose the matrix into an orthogonal matrix and a triangular matrix. \) The Cholesky algorithm, used to calculate the decomposition matrix L, is a modified version of Gaussian elimination. It computes an incomplete factorization of the coefficient matrix and . So as the the rest processors inner-most loop, $T_2 = \frac{O(b^3)+S(b^2)}{N_p}$ . Those are colored in light blue. It is easy to observe that the left-looking algorithm has $A_{jj}$ subtract all is left sub-matrixes square. Algorithm for Cholesky Decomposition Input: an nn SPD matrix A Output: the Cholesky factor, a lower triangular matrix L such that A = LLT Theorem:(proof omitted) For a symmetric matrix A, the Cholesky algorithm will succeed with non-zero diagonal entries in L if and only if A is SPD. General-purpose graphics processing units (GPGPUs) could bring huge performance improvements in scientific and numerical fields. 2 (c) we show how minor modifications Hello I am trying to implement the following algorithm for Cholesky Decomposition Column-Wise Method: for j=1:n for i=1:j-1 end end My attempt so far at implementing the above: Theme Copy A= [4 -1 1; -1 4.25 2.75; 1 2.75 16;]; % Check R matches with col (A); count = 0; [n,n] = size (A); R=zeros (n,n) for j=1:n for i=1:j-1 sum1 = 0 for k=1:i-1 2 (b) is a straight large number of communications. /Length 2288 We survey the literature and determine which of the existing modi ed Cholesky algorithms is most suitable for inclusion in the Numerical Algorithms Group >> part of the matrix that still needs to be factored ( in If matrix M is nonsingular, then this factorization or decomposition is unique. the above algorithm has the effect that after k Algorithm for Cholesky Factorization for a Hermitian positive def-inite matrix Step1. Similarity in the derivation First, we analyzed the implementation of Cholesky factorization in MAGMA and identified the bottleneck of the current implementation, which is the use of fixed block size without . If that made zero sense, this is how it looks: . Cholesky Decomposition Definition 1: A matrix A has a Cholesky Decomposition if there is a lower triangular matrix L all whose diagonal elements are positive such that A = LLT. is also rank-factorization of Transpose of M. Let M = PQ where P is a (m,k) and Q is (k,n) dimensions. For example for a matrix with non-zeros only along the first row, first column, and diagonal the Cholesky factors have 100% fill-in (the lower and upper triangles are 100% dense). For a given block size r, we can write. to the PLAPACK implementation in (b) allows us to force to exist on one node. It is closely connected with the solution of least-squares problems (cf. If you are going through hell, keep going. If pivoting is used, then two additional attributes "pivot" and "rank" are also returned. So in in Addition to the correct answer of @Avi you can also use the right class for standard cholesky decomposition: Eigen::LLT<MatrixXd> tmp (matA); cout << MatrixXd (tmp.matrixU ()) << "\n\n"; Share Follow Decomposition plays vital role in algorithms and easily compute various types of matrices and can work on specific elements in the Matrices. The upper triangular factor of the Cholesky decomposition, i.e., the matrix R such that R'R = x (see example). By the Cholesky decomposition, for a symmetric positive define matrix , we have, \(L_{jj} = chol( A _{ij} )\)\(L_{ij}=(A_{ij} - \sum_{k=1}^{j-1}L_{ik}L_{jk}^T)L_{jj}^{-T}\). steps. I choose this to preserve a consistent program behavior. Given a symmetric positive definite matrix , the Cholesky decomposition is an upper triangular matrix with strictly positive diagonal entries such that. of nodes. mxn calc. As a result time for round j is: $\sum_{i=j+1}^{N_b-1}\sum_{k=j+1}^{i+1} \max (T_1, T_2)$. As a result, We could insert the laplace operator These sequential can be easily re-written with SIMD or OpenMP for better intra-node parallelism. Taking both into account, round time j is $\max(T_3,T_4)$. The most useful of these is the Singular Value Decomposition. Various versions of level-3 BLAS based Cholesky factorization using Cholesky Factorization (no JIT) Cholesky Factorization (JIT) Gussian Elimination Cholesky (no JIT) vs Cholesky (JIT) Cholesky Factorization VS Gaussian Elimination I use additional loop to send $A_{jk}$ earlier instead of send $A_{jk} and update $A_{jj}$ in the same loop. numpy.linalg.cholesky# linalg. The primary reason for this is that the duplicate the data as part of the parallel To illustrate, we describe a partitioned Cholesky factorization algorithm. The remainder of the code is self Below is the comparison with single-node and 2-node execution with the same data size: To solve this problem, just replace the basic block Cholesky decomposition and matrix multiplication & inversion & subtraction to in-node parallelism versions implemented by OpenMP and SIMD, but I havent implement them in this report. Properties of eigenvalues and eigenvectors: Definite/Semi-definite Matrices Based on Eigen decomposition: Pros: Once you apply eigen decomposition on square matrix then you will get other properties very easily like trace, determinant, rank, diagonals, etc.. Cons: Eigen Decomposition works only on Square Matrices. Passionate in Autonomous. The resulting PLAPACK code is given in Fig. The right-looking almost does not scale. also Least squares, method of ), since the normal equations that characterize the least-squares solution have a symmetric positive-definite coefficient matrix. The most popular and widely used algorithms which are based on Decomposition methods are PCA and SVD, which are pure Linear Algebra based. For a given block size r, we can write. Naturally, one may get the impression that we have merely in Fig. so we seek methods that take advantage of the special properties Powered by Jekyll & So Simple. In some cases it is convenient to rewrite this decomposition in its equivalent form , where is an upper triangular matrix. For scalability reasons, parallel dense linear algebra And, unlike the LU factorization, the Cholesky factorization is simply backward stable | no appeal to pivot growth factors is required. The best example for application of SVD , PCA is Image Compression in Computer Vision. matrix to be factored. The size of data here is a square matrix of 16000 dimensions, which is chosen to run both algorithms in 15min. Decomposition allows us to Decompose the Matrix in the form of vectors and matrices. code in (b) closely. based) algorithm, as is shown in Fig. Cholesky Decomposition and Linear Programming on a GPU if X is real and I is the identity matrix. The strong scalability test of both decomposition algorithms are conducted only on a single node of Oakbridge-CX. ! It can be often applied to various matrices like Square matrix and Rectangular matrix. operations on data. Second, we compare the cost of various Cholesky decomposition implementations to this lower bound, and draw the following conclusions: (1) "Nave" sequential algorithms for Cholesky attain nei-ther the bandwidth nor latency lower bounds. \(\sum_{j=0}^{N_b-1}\sum_{i=j+1}^{N_b-1}\sum_{k=j+1}^{i+1} 2 = \frac{N_b^{3}-3N_b^2-4N_b}{3}\) A QR decomposition of a real square matrix M is a decomposition of M as. natural intermediate distribution for the data This function will be called in the main body of all processes. 3 (October 2008): 1-14. I analyze operations by every round computing column j. on the panel is .Thus, if the block size b is relatively large, Find a LU decomposition of A = LU. An incomplete Cholesky factorization is given by a sparse lower triangular matrix K that is in some sense close to L. % Factorize A such that A = L*L', % where L is a lower triangular matrix whose diagonal entries are not. and R is an upper triangular matrix. For processor having column j, suppose send blocks until message delivered for simplicity, the time for one inner-most loop is $T_1 = (\frac{O(b^3)}{N_p}+\frac{S(b^2)(N_p-1)}{N_p})$ , where $O(b^3)$ is matrix multiplication (which can be optimized further) for a $b \times b$ block, and $S(b^2)$ is time to send such a block. What is important is that these blocks are assigned When it is applicable, the Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems of linear equations. Cholesky decomposition is the dominant calculation where T the next three parts are relatively simple due to the triangular structure of the matrices G and G . exists within one column of nodes. This factorization exists and is unique for positive definite matrices. 1x$pSH7"TyJ hgpFm$/ Oq,$xN1!1"Vw"Wpy\qbs[z~;qo]e F0 1/F+!\50!7h)>>M;0 /f5~%8RXd~-FMx 7u`4L(5l+a9 k}F^YD|9,(A5!fs2V+`xU]*^T~He_aJ(5h9Q@RI$z?As b7Azym"pUifX#U-q05mz9 -5uD$x>gu d;:BgQj3u RyU;d?IbDL{.KOaDopUW`}W&t&LK}e_2mA=C{7ahiC|RC.84!q#z2C~ "f5%D?BfuArHxK+ RZx;k>V"+%S)y>WZ:d?>m{(WGAm*GIiI"?U-: [ Sometimes they are polynomials such as Z-transforms. Fig. Subsequently, we calculate the off-diagonals for the elements below the diagonal: substantially more complex than the examples presented. To understand further optimizations, one once again needs to know entirely on one node. Viola Jones Algorithm. factorization. data volume ratio, allowing for more effective use Taking matrix length 24288 and number of CPU 56 as an example, below figures demonstrates the performance difference: in the critical path, and thus contributes considerably Seems the left-looking algorithm scales well in a single node. is , where r equals the MX = 0 => X =0 . For understanding the code, the sizes of the blocks are not important. so that hidden all complexity and ugliness in routines like solving simultaneous equations there is an advantage to redistributing the panel. If (B,C) is a rank-factorization of M, then Transport of B, C i.e., P is of full column rank and Q is of full row rank. The factorization is named after A.-L. Cholesky, a French military officer involved in geodesy. One solution I'm aware is to find a permutation P matrix and do the Cholesky decomposition of PTAP. To illustrate, we describe a partitioned Cholesky factorization algorithm. xY[~0$5wRfsilhZ['VN%;$,9O $3Xp'2_l_?+4W0b9be5gjC}JZipkb3[s,Jq5Kt+w/>WKcJdGzVc?/XN/fJ[IY7PK+f\I$HKc2.EDb%ae/CY/1?lp:m|z:QFq{*_v,k/xrHmimw_RlHv}^zHJt GvD|$c9 1(+!M~- NnOs92's&$wA_vCnHzJneFGe{IKD 8[H|Bp"|7"sVU@>I(B"sE$XU3>Qr>@Mp(mP" *U/U @ 6y2 /W;R&Zf">!G#EYi Every non-null matrix has a rank-factorization. The paper talks about an implementation of the Cholesky decomposition, modified by the fact that when the matrix is not positive definite, a diagonal perturbation is added before the decomposition takes place. into the polynomial division program, and then explicitly multiply these elements Assignment of matrices is then the number of columns that can be split off from the left of, the number of rows that can be split off from the top of. call to a basic (nonblocked) implementation of the Cholesky The parallel program is tested on a single node since they dont scale when network communication introduced. Equating (i,j) elements in the equation i.e.. This operation is $O(n^3)$: Matrix-matrix multiplication only happens as the form of $AB^T$ , which is cache friendly: All these 3 algorithms exploit the symmetry of matrix to reduce computation. If one views the panel of the matrix Factor U = D2W where W is a unit upper-triangular matrix and D is a diagonal matrix. Poisson's equation {? bx } .Qzk%SyE 2, The QZ decomposition is also called Generalized Schur Decomposition where S and T are the Schur form of A and B matrices. Assignment of a vector to the nodes is Cholesky Factorization Algorithm Scrivere qualcosa di carino Risultati I tempi di esecuzione sono espressi in secondi (s). and because the Cholesky method Suppose row 1 $(A_{11} , A_{22}, A_{33})$ has been computed, then Most time spent on right-looking is communication. an initial-value problem. The following number of operations should be performed to decompose a matrix of order n using a serial version of the Cholesky algorithm: n square roots n ( n 1) 2 divisions n 3 n 6 multiplications and n 3 n 6 additions (subtractions): the main amount of computational work. The Cholesky algorithm, used to calculate the decomposition matrix L, is a modified version of Gaussian elimination. The inner loop computation of matrix multiplication and matrix subtract is paralleled on different processors. a further optimization is attained by duplicating within .To find Unitary means. Figure 1: The answer is to look into the correlation between the points. What are the limitations of the Cholesky decomposition algorithm? stream 2 (c). We propose an approximation to the forward filter backward sampler (FFBS) algorithm for large-scale spatio-temporal smoothing. when we filter starting from both ends. /Filter /FlateDecode (and thus ), I've written this algorithm as matlab function which accepts one parameter as Matrix for Cholesky decomposition. The slope of left looking is close to 1 but slightly goes down when the number of cores is more than 40. It is this ratio of computation to data volume (requiring Decomposition can be done for both Square and Non-square matrices. Below is the data dependency, the red cell lying at diagonal depends on all its left cells(green). The Cholesky decomposition is mainly used for the numerical solution of linear equations \( {\bf A}{\bf x} = {\bf b} . a multivector and, more importantly, considerably There are several methods exists for computing QR decomposition. a parallel matrix-vector based routine, which requires an extremely A further optimization of the parallel Cholesky Given the derivation of the First, we calculate the values for L on the main diagonal. We presented two approaches utilizing hybrid CPU/GPU system in Cholesky factorization. % necessarily unity. Cholesky factor Any symmetric positive definite matrix can be factored as where is lower triangular matrix. a two-dimensional cartesian distribution: Given that the currently active submatrix is distributed to A variant of Cholesky decomposition is the factorization in Lower triangular and D is diagonal. It is defined as follows: It is also called generalized Schur Decomposition. we first backsolve Here, the full matrix is partitioned into smaller sub-matrixes, and bellow is an example of partition A in to 9 sub-matrixes: The sub-matrixes are distributed by block row/column cyclic data distribution. It is applicable to Square matrices and can be defined as follows. For left-looking, the parallel execution demonstrated as following: Here the broadcast in left process will not block So total time for round j is to reformulate the algorithm in terms of .When The Cholesky algorithm takes a positive-definite matrix and factors it into a triangular matrix times its transpose, say . Edit @chtz is right - using Upper wont give you the result you expect because LDLT class is for robust cholesky decomposition with pivoting. decomposition use either equation solving [3] or triangular matrix operations [4] with most efficient implementation If , with is the . to the overall time required for the Cholesky factorization. FFBS is commonly used in Bayesian statistics when working with linear Gaussian state-space models, but it requires inverting covariance matrices which have the size of the latent state vector. No checking is performed to verify whether a is . An incomplete Cholesky factorization is often used as a preconditioner for algorithms like the conjugate gradient method . A_{22} = chol( A _{22} )\\ are based on rewriting the equation \( {\bf A} = {\bf L}{\bf L}^{\ast} \) as \[ Recognizing that is a row-wise operation and thus parallelizes trivially Computing the factorization can be sped up for a few specific patterns such as trees, but no algorithm has been shown to work in a generic way for SDD Let M be a matrix of dimensions(m,n) with rank 1. In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / lski / sh-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations. Decomposition is very important in algorithmic perspective in order to understand performance. The parallel program with parallel degree 1 still divides the whole matrix into sub-matrixes.

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