Matrix splitting

In the mathematical discipline of numerical linear algebra, a matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. Many iterative methods (for example, for systems of differential equations) depend upon the direct solution of matrix equations involving matrices more general than tridiagonal matrices. These matrix equations can often be solved directly and efficiently when written as a matrix splitting. The technique was devised by Richard S. Varga in 1960. A x = k , {displaystyle mathbf {A} mathbf {x} =mathbf {k} ,}     (1) A = B − C , {displaystyle mathbf {A} =mathbf {B} -mathbf {C} ,}     (2) B x = g , {displaystyle mathbf {B} mathbf {x} =mathbf {g} ,}     (3) B x ( m + 1 ) = C x ( m ) + k , m = 0 , 1 , 2 , … , {displaystyle mathbf {B} mathbf {x} ^{(m+1)}=mathbf {C} mathbf {x} ^{(m)}+mathbf {k} ,quad m=0,1,2,ldots ,}     (4) x ( m + 1 ) = B − 1 C x ( m ) + B − 1 k , m = 0 , 1 , 2 , … {displaystyle mathbf {x} ^{(m+1)}=mathbf {B} ^{-1}mathbf {C} mathbf {x} ^{(m)}+mathbf {B} ^{-1}mathbf {k} ,quad m=0,1,2,ldots }     (5) A = D − U − L , {displaystyle mathbf {A} =mathbf {D} -mathbf {U} -mathbf {L} ,}     (6) x ( m + 1 ) = D − 1 ( U + L ) x ( m ) + D − 1 k . {displaystyle mathbf {x} ^{(m+1)}=mathbf {D} ^{-1}(mathbf {U} +mathbf {L} )mathbf {x} ^{(m)}+mathbf {D} ^{-1}mathbf {k} .}     (7) x ( m + 1 ) = ( D − L ) − 1 U x ( m ) + ( D − L ) − 1 k . {displaystyle mathbf {x} ^{(m+1)}=(mathbf {D} -mathbf {L} )^{-1}mathbf {U} mathbf {x} ^{(m)}+(mathbf {D} -mathbf {L} )^{-1}mathbf {k} .}     (8) x ( m + 1 ) = ( D − ω L ) − 1 [ ( 1 − ω ) D + ω U ] x ( m ) + ω ( D − ω L ) − 1 k . {displaystyle mathbf {x} ^{(m+1)}=(mathbf {D} -omega mathbf {L} )^{-1}mathbf {x} ^{(m)}+omega (mathbf {D} -omega mathbf {L} )^{-1}mathbf {k} .}     (9) A = ( 6 − 2 − 3 − 1 4 − 2 − 3 − 1 5 ) , k = ( 5 − 12 10 ) . {displaystyle mathbf {A} ={egin{pmatrix}6&-2&-3\-1&4&-2\-3&-1&5end{pmatrix}},quad mathbf {k} ={egin{pmatrix}5\-12\10end{pmatrix}}.}     (10) B = ( 6 0 0 0 4 0 0 0 5 ) , C = ( 0 2 3 1 0 2 3 1 0 ) , {displaystyle {egin{aligned}&mathbf {B} ={egin{pmatrix}6&0&0\0&4&0\0&0&5end{pmatrix}},quad mathbf {C} ={egin{pmatrix}0&2&3\1&0&2\3&1&0end{pmatrix}},end{aligned}}}     (11) x ( m + 1 ) = ( 0 1 3 1 2 1 4 0 1 2 3 5 1 5 0 ) x ( m ) + ( 5 6 − 3 2 ) , m = 0 , 1 , 2 , … {displaystyle mathbf {x} ^{(m+1)}={egin{pmatrix}0&{frac {1}{3}}&{frac {1}{2}}\{frac {1}{4}}&0&{frac {1}{2}}\{frac {3}{5}}&{frac {1}{5}}&0end{pmatrix}}mathbf {x} ^{(m)}+{egin{pmatrix}{frac {5}{6}}\-3\2end{pmatrix}},quad m=0,1,2,ldots }     (12) x = ( 2 − 1 3 ) . {displaystyle mathbf {x} ={egin{pmatrix}2\-1\3end{pmatrix}}.}     (13) D = ( 6 0 0 0 4 0 0 0 5 ) , U = ( 0 2 3 0 0 2 0 0 0 ) , L = ( 0 0 0 1 0 0 3 1 0 ) . {displaystyle mathbf {D} ={egin{pmatrix}6&0&0\0&4&0\0&0&5end{pmatrix}},quad mathbf {U} ={egin{pmatrix}0&2&3\0&0&2\0&0&0end{pmatrix}},quad mathbf {L} ={egin{pmatrix}0&0&0\1&0&0\3&1&0end{pmatrix}}.}     (14) x ( m + 1 ) = 1 120 ( 0 40 60 0 10 75 0 26 51 ) x ( m ) + 1 120 ( 100 − 335 233 ) , m = 0 , 1 , 2 , … {displaystyle mathbf {x} ^{(m+1)}={frac {1}{120}}{egin{pmatrix}0&40&60\0&10&75\0&26&51end{pmatrix}}mathbf {x} ^{(m)}+{frac {1}{120}}{egin{pmatrix}100\-335\233end{pmatrix}},quad m=0,1,2,ldots }     (15) x ( m + 1 ) = 1 12 ( − 1.2 4.4 6.6 − 0.33 0.01 8.415 − 0.8646 2.9062 5.0073 ) x ( m ) + 1 12 ( 11 − 36.575 25.6135 ) , m = 0 , 1 , 2 , … {displaystyle mathbf {x} ^{(m+1)}={frac {1}{12}}{egin{pmatrix}-1.2&4.4&6.6\-0.33&0.01&8.415\-0.8646&2.9062&5.0073end{pmatrix}}mathbf {x} ^{(m)}+{frac {1}{12}}{egin{pmatrix}11\-36.575\25.6135end{pmatrix}},quad m=0,1,2,ldots }     (16) In the mathematical discipline of numerical linear algebra, a matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. Many iterative methods (for example, for systems of differential equations) depend upon the direct solution of matrix equations involving matrices more general than tridiagonal matrices. These matrix equations can often be solved directly and efficiently when written as a matrix splitting. The technique was devised by Richard S. Varga in 1960. We seek to solve the matrix equation where A is a given n × n non-singular matrix, and k is a given column vector with n components. We split the matrix A into where B and C are n × n matrices. If, for an arbitrary n × n matrix M, M has nonnegative entries, we write M ≥ 0. If M has only positive entries, we write M > 0. Similarly, if the matrix M1 − M2 has nonnegative entries, we write M1 ≥ M2. Definition: A = B − C is a regular splitting of A if B−1 ≥ 0 and C ≥ 0.