aa r X i v : . [ c s . S C ] F e b Fast generalized Bruhat decomposition
Gennadi Malaschonok ⋆ Tambov State University,Internatsionalnaya 33, 392622 Tambov, Russia
Abstract.
The deterministic recursive pivot-free algorithms for the com-putation of generalized Bruhat decomposition of the matrix in the fieldand for the computation of the inverse matrix are presented. This methodhas the same complexity as algorithm of matrix multiplication and it issuitable for the parallel computer systems. An LU matrix decomposition without pivoting is a decomposition of the form A = LU , a decomposition with partial pivoting has the form P A = LU , anddecomposition with full pivoting (Trefethen and Bau) has the form P AQ = LU , where L and U are lower and upper triangular matrices, P and Q is apermutation matrix.French mathematician Francois Georges Ren´e Bruhat was the first who workedwith matrix decomposition in the form A = V wU , where V and U are nonsin-gular upper triangular matrices and w is a matrix of permutation. Bruhat de-composition plays an important role in algebraic group. The generalized Bruhatdecomposition was introduced and developed by D.Grigoriev[1],[2]. He uses theBruhat decomposition in the form A = V wU , where V and U are upper trian-gular matrices but they may be singular when the matrix A is singular. In thepapers [3] and [4] there was analyzed the sparsity pattern of triangular factorsof the Bruhat decomposition of a nonsingular matrix over a field.Fast matrix multiplication and fast block matrix inversion were discovered byStrassen [5]. The complexity of Strassen’s recursive algorithm for block matrixinversion is the same as the complexity of an algorithm for matrix multiplica-tion. But in this algorithm it is assumed that principal minors are invertible andleading elements are nonzero as in the most of direct algorithms for matrix in-version. There are known other recursive methods for adjoint and inverse matrixcomputation, which have the complexity of matrix multiplications([6]-[8]).In a general case it is necessary to find suitable nonzero elements and toperform permutations of matrix columns or rows. Bunch and Hopkroft suggestedsuch algorithm with full pivoting for matrix inversion [9]. ⋆ Preprint of the paper: G. Malaschonok. Fast Generalized Bruhat Decomposition.Computer Algebra in Scientific Computing, LNCS 6244, Springer, Berlin, 2010.P. 194-202. (Supported by the Sci. Program Devel. Sci. Potent. High. School,RNP.2.1.1.1853.) Gennadi Malaschonok
The permutation operation is not a very difficult operation in the case of se-quential computations by one processor, but it is a difficult operation in the caseof parallel computations, when different blocks of a matrix are disposed in differ-ent processors. A matrix decomposition without permutations is needed for par-allel computation for construction of efficient and fast computational schemes.The problem of obtaining pivot-free algorithm was studied in [10],[11] byS.Watt. He presented the algorithm that is based on the following identity fora nonsingular matrix: A − = ( A T A ) − A T . Here A T is the transposed matrixto A and all principal minors of the matrix A T A are nonzero. This method isuseful for making an efficient parallel program with the help of Strassen’s fastdecomposition of inverse matrix for dense nonsingular matrix over the field ofzero characteristic when field elements are represented by the float numbers.Other parallel matrix algorithms are developed in [12] - [15].This paper is devoted to the construction of the pivot-free matrix decompo-sition method in a common case of singular matrices over a field of arbitrarycharacteristic. The decomposition will be constructed in the form LAU = E ,where L and U are lower and upper triangular matrices, E is a truncated per-mutation matrix, which has the same rank as the matrix A . Then the generalizedBruhat decomposition may be easy obtained using the matrix L , E and U . Thisalgorithm has the same complexity as matrix multiplication and does not re-quire pivoting. For singular matrices it allows to obtain a nonsingular block ofthe biggest size, the echelon form and kernel of matrix. The preliminary variantsof this algorithm were developed in [16] and [17]. We introduce some notations that will be used in the following sections.Let F be a field, F n × n be an n × n matrix ring over F , S n be a permutationgroup of n elements. Let P n be a multiplicative semigroup in F n × n consisting ofmatrices A having exactly rank(A) nonzero entries, all of them equal to 1. Wecall P n the permutation semigroup because it contains the permutation groupof n elements S n and all their truncated matrix.The semigroup D n ⊂ P n is formed by the diagonal matrices. So | D n | =2 n and the identity matrix I is the identity element in D n , S n and P n .Let W i,j ∈ P n be a matrix, which has only one nonzero element in the position( i, j ). For an arbitrary matrix E of P n , which has the rank n − s ( s = 0 , ..n )we shall denote by i E = { i , .., i s } the ordered set of zero row numbers and i E = { i , .., i s } the ordered set of zero column numbers. Definition 1.
Let E ∈ P n be the matrix of the rank n − s , let i E = { i , .., i s } and i E = { i , .., i s } are the ordered set of zero row numbers and zero columnsnumber of the matrix E . Let us denote by E the matrix E = X k =1 ,..s W i k ,j k and call it the complimentary matrix for E . For the case s = 0 we put E = 0 . ast generalized Bruhat decomposition 3 It is easy to see that ∀ E ∈ P n : E + E ∈ S n , and ∀ I ∈ D n : I + I = I .Therefore the map I I = I − I is the involution and we have II = 0. We candefine the partial order at D n : I < J ⇔ J − I ∈ D n . For each matrix E ∈ P n we shall denote by I E = EE T and J E = E T E the diagonal matrix: I E , J E ∈ D n . The unit elements of the matrix I E shownonzero rows of the matrix E and the unit elements of the matrix J E shownonzero columns of the matrix E . Therefore we have several zero identities: E T I E = I E E = EJ E = J E E T = 0 . (1)For any pare I, J ∈ D n let us denote the subset of matrices F n × n F n × nI,J = { B : B ∈ F n × n , IBJ = B } . We call them (
I, J )-zero matrix. It is evident that F n × n = F n × n I , I , 0 ∈ ∪ I,J F n × nI,J and if I < I and J < J then F n × nI ,J ⊂ F n × nI ,J . Definition 2.
We shall call the factorization of the matrix A ∈ F n × nI,J A = L − EU − , (2) LEU -decomposition if E ∈ P n , L is a nonsingular low triangular matrix, U isan upper unitriangular matrices and L − I E ∈ F n × nI,I E , U − J E ∈ F n × nJ E ,J . (3)If (2) is the LEU -decomposition we shall write(
L, E, U ) = LU ( A ) , Sentence 1
Let ( L, E, U ) = LU ( A ) be the LEU -decomposition of matrix A ∈ F n × nI,J then L = I E + ILI E , U = J E + J E U J, E ∈ F n × nI,J , (4) L − = I E + L − I E , U − = J E + J E U − . Proof.
The first and second equalities follows from (3). To prove the property ofmatrix E we use the commutativity of diagonal semigroup D n : E = LAU = ( I E + ILI E ) IAJ ( J E + J E U J ) = I ( I E + LI E I ) A ( J E + JJ E U ) J. To prove the property of matrix L − let us consider the identity I = L − L = L − ( I E + LI E ) = L − I E + I I E Therefore L − I E = I E and L − = L − ( I E + I E ) = I E + L − I E . The prove ofthe matrix U − property may be obtained similarly. Gennadi Malaschonok
Sentence 1 states the property of matrix E , which may be written in the form I E < I , J E < J . We shall call it the property of immersion . on the other hand,each zero row of the matrix E goes to the unit column of matrix L and eachzero column of the matrix E goes to the unit row of matrix U .Let us denote by E n the permutation matrix W ,n + W ,n − + .. + W n, ∈ S n .It is easy to see that if the matrix A ∈ F n × n is low-(upper-) triangular, then thematrix E n A E n is upper- (low-) triangular. Sentence 2
Let ( L, E, U ) = LU ( A ) be the LEU -decomposition of matrix A ∈ F n × n , then the matrix E n A has the generalized Bruhat decomposition V wV and V = E n ( L − − I E ) E n , w = E n ( E + E ) , V = ( U − − J E ) . Proof.
As far as L − is a low triangular matrix and U − is an upper triangularmatrix we see that V and V are upper triangular matrices. Matrix w is a productof permutation matrices so w is a permutation matrix. One easily checks that V wV = E n L − EU − = E n A .Examples.For any matrix I ∈ D n , E ∈ P n , 0 = a ∈ F the product ( aI + I ) I I isa LEU decompositions of matrix aI and the product ( aI E + I E ) E I is a LEUdecompositions of matrix aE . LEU decomposition
Theorem 1.
For any matrix A ∈ F n × n of size n = 2 k , k ≥ a LEU -decomposition exists. For computing such decomposition it is enough to com-pute 4
LEU -decompositions, 17 multiplications and several permutations for thematrices of size n = 2 k − .Proof. For the matrix of size 1 ×
1, when k = 0, we can write the following LEU decompositions LU (0) = (1 , ,
1) and LU ( a ) = ( a − , , , if a = 0 . Let us assume that for any matrix of size n we can write a LEU decompositionand let us given matrix A ∈ F n × nI,J has the size 2 n . We shall construct a LEU decomposition of matrix A .First of all we shall divide the matrices A , I , J and a desired matrix E intofour equal blocks: A = (cid:20) A A A A (cid:21) , I = diag( I , I ) , J = diag( J , J ) , E = (cid:20) E E E E (cid:21) , (5)and denote I ij = E ij E Tij , J ij = E Tij E ij ∀ i, j ∈ { , } . (6)Let ( L , E , U ) = LU ( A ) , (7) ast generalized Bruhat decomposition 5 denote the matrices Q = L A , B = A U , (8) A = BJ , A = I Q, A = A − BE T Q. (9)Let ( L , E , U ) = LU ( A ) and ( L , E , U ) = LU ( A ) , (10)denote the matrices G = L A U , A = I GJ . (11)Let us put ( L , E , U ) = LU ( A ) , (12)and denote W = ( GE T L + L BE T ) , V = ( U E T GJ + E T QU ) , (13) L = (cid:18) L L − L W L L L (cid:19) , U = (cid:18) U U − U V U U U (cid:19) . (14)We have to prove that ( L, E, U ) = LU ( A ) . (15)As far as L , L , L , L are low triangular nonsingular matrices and U , U , U , U are upper unitriangular matrices we can see in (10) that the matrix L is a low triangular nonsingular matrix and the matrix U is upper unitriangular.Let us show that E ∈ P n . As far as E , E , E , E ∈ P n and A = I A J , A = BJ , A = I Q , A = I GJ and due to the Sentence 1we obtain E = I E J , E = E J , E = I E , E = I E J .Therefore the unit elements in each of the four blocks of the matrix E aredisposed in different rows and columns of the matrix E . So E ∈ P n , and nextidentities hold E E T = E J = J E T = J J = 0 , (16) E T E = E T I = I E = I I = 0 , (17) E E T = E J = J E T = J J = 0 , (18) E T E = E T I = I E = I I = 0 . (19)We have to prove, that E = LAU . This equation in block form consists offour block equalities: E = L L A U U ; E = L L ( A U − A U V ) U ; E = L ( L A − W L A ) U U ; E = L (( L A − W L A ) U − ( L A − W L A ) U V ) U . (20)Therefore we have to prove these block equalities. Gennadi Malaschonok
Let us note, that from the identity A = I A J and Sentence 1 we get L = I + I L I , U = J + J U J . (21)The Sentence 1 together with equations A = I L A , A = A U J , A = I L ( A − A U E T L A ) U J give the next properties of L- andU- blocks: L = I + I I L I , U = J + J U J ,L = I + I L I , U = J + J U J J ,L = I + I I L I , U = J + J U J J . (22)The following identities can be easy checked now L E = E , L I = I , (23) E U = E , J U = J , (24) E U = E , J U = J , (25) L E = E , L I = I . (26)We shall use the following equalities, L A U = E , L A U = E , L A U = E , L A U = E , (27)which follows from (7),(10) and (12), the equality E V = I QU , (28)which follows from the definition of the block V in (13), (24), (16) and (6), theequality W E = L BJ , (29)which follows from the definition of the block W in (13), (23), (17) and (6).1. The first equality of (20) follows from (27), (23) and (24).2.The right part of the second equality of (20) takes the form L ( I − I ) QU U due to (8), (27) and (28). To prove the second equality we usethe definition of the blocks B and A in (8) and (9), then the second equality in(27) and identity (25): L ( I − I ) QU U = L A U U = E U = E .
3. The right part of the third equality of (20) takes the form L L B ( I − J ) U due to definition of the block B (8), the first equality in (27) and (29).To prove the third equality we use the definition of the blocks A in (9), thenthe third equality in (27) and identity (26): L L BJ U = L L A U = L E = E .4. The identity E T L = E T L ( I + I ) = E T L I (30)follows from (23) and (17). ast generalized Bruhat decomposition 7 We have to check that ( L A − W L A ) U = ( L A − ( GE T L + L BE T ) Q ) U = L ( A − BE T Q ) U − GE T L QU = L A U − GE T L I QU = G − GE T L A U = G − GE T E = GJ , using thedefinitions of the blocks W in (13), A and A in (9), the identity (28), thesecond equality in (27) and the definition (6).We have to check that − ( L A − W L A ) U V = − ( L A U − W E ) V = ( − L B + L BJ ) V = − L BJ V = − L BJ ( U E T GJ + E T QU ) = − L A U E T GJ = − I GJ , using the first equality in (27), the identity(29), the definitions of the blocks V in (13), (1), then the third equality in (27)and definition (6).To prove the forth equality we have to substitute obtained expressions to theright part of the fourth equality: L ( GJ − I GJ ) U = L I GJ U = L A U = E . For the completion of the proving of this theorem we have to demonstratethe special form of the matrices U and L : L − I E ∈ F I,I E and U − J E ∈ F J E ,J .The matrix L is invertible and I E < I therefore we have to prove that L = I E + ILI E , where I E = diag( I + I , I + I ), I E = diag( I I , I I ), I = diag( I , I ).This matrix equality for matrix L (14) is equivalent to the four block equal-ities: L L = I L L ( I + I ) + I I , I I + I ) , − L W L = − I L W L ( I + I ) , L L = I L L ( I + I ) + I I . To prove the first block equalities we have to multiply its left part by the unitmatrix in the form I = ( I + I ) from the left side and by the unit matrix inthe form I = ( I + I ) + I I from the left side. Then we use the followingidentities to obtain in the left part the same expression as in the right part: L I = I , L I = I , I L L = I , I ( I + I ) = 0. The same ideamay be used for proving the last block equality, but we must use other forms ofunit matrix: I = ( I + I ), I = ( I + I ) + I I .The second block equality is evident.Let us prove the third block equality. We have to multiply the left part ofthe third block equality by the unit matrix in the form I = ( I + I ) from theleft side and by the unit matrix in the form I = ( I + I ) + I I from theright side.The block W is equal to the following expression by the definition (13), (11)and (8): W = ( L ( A − A U E T Q ) U E T L + L A U E T ) . We have to use in the left part the equations I L = I , I L = I , I A = 0, I A = 0, and L I = I , L I = I , E T I = 0, E T I = 0.The property of the matrix U : U − J E ∈ F J E ,J may be proved in the sameway as the property of the matrix L . Gennadi Malaschonok
Theorem 2.
For any matrix A of size s, ( s ≥ , an algorithm of LEU -decompo-sition which has the same complexity as matrix multiplication exists.Proof.
We have proved an existence of
LEU -decomposition for matrices of size2 k , k > . Let A ∈ F s × sI,J be a matrix of size 2 k − < s < k , A ′ be a matrixof size 2 k , which has in the left upper corner the submatrix equal A and allother elements equal zero. We can construct LEU -decomposition of matrix A ′ :( L ′ , E ′ , U ′ ) = LU ( A ′ ). According to the Sentence 1 the product L ′ A ′ U ′ = E ′ has the form (cid:18) L I (cid:19) (cid:18) A
00 0 (cid:19) (cid:18) U I (cid:19) = (cid:18) E
00 0 (cid:19)
Therefore
LAU = E is a LEU decomposition of matrix A .The total amount of matrix multiplications in (7)-(15) is equal to 17 andtotal amount of recursive calls is equal to 4. We do not consider multiplicationsof the permutation matrices, we can do these multiplications due to permutationof pointers for the blocks which are disposed at the lockal processors.We can compute the decomposition of the second order matrix by means of5 multiplicative operations. Therefore we get the following recurrent equality forcomplexity t ( n ) = 4 t ( n/
2) + 17 M ( n/ , t (2) = 5 . Let γ and β be constants, 3 ≥ β >
2, and let M ( n ) = γn β + o ( n β ) be the numberof multiplication operations in one n × n matrix multiplication.After summation from n = 2 k to 2 we obtain17 γ (4 β ( k − + . . . + 4 k − β ) + 4 k − γ n β − β − n β − n . Therefore the complexity of the decomposition is ∼ γn β β − A is an invertible matrix, then A − = U E T L and a recursive block algo-rithm of matrix inversion is written in the expressions (7)-(15). This algorithmhas the complexity of matrix multiplications. An algorithms for finding the generalized Bruhat decomposition and matrixinversion are described. These algorithms have the same complexity as matrixmultiplication and do not require pivoting. For singular matrices they allow toobtain a nonsingular block of the biggest size. These algorithms may be used inany field, including real and complex numbers, finite fields and their extensions.The proposed algorithms are pivot-free, and do not change the matrix blockstructure. So they are suitable for parallel hardware implementation. ast generalized Bruhat decomposition 9
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