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| In the [[mathematics|mathematical]] discipline of [[numerical linear algebra]], a '''matrix splitting''' is an expression which represents a given [[matrix (mathematics)|matrix]] as a sum or difference of matrices. Many [[iterative method]]s (e.g., for systems of [[differential equation]]s) depend upon the direct solution of matrix equations involving matrices more general than [[tridiagonal matrix|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.<ref>{{harvtxt|Varga|1960}}</ref>
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| | |
| ==Regular splittings==
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| We seek to solve the [[Matrix(mathematics)#Linear_equations|matrix equation]]
| |
| | |
| : <math> \bold A \bold x = \bold k, \quad (1) </math>
| |
| | |
| where '''A''' is a given ''n'' × ''n'' [[invertible matrix|non-singular]] matrix, and '''k''' is a given [[column vector]] with ''n'' components. We split the matrix '''A''' into
| |
| | |
| :<math> \bold A = \bold B - \bold C, \quad (2) </math>
| |
| | |
| 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 '''M'''<sub>1</sub> − '''M'''<sub>2</sub> has nonnegative entries, we write '''M'''<sub>1</sub> ≥ '''M'''<sub>2</sub>.
| |
| | |
| Definition: '''A''' = '''B''' − '''C''' is a '''regular splitting of A''' if and only if '''B'''<sup>−1</sup> ≥ '''0''' and '''C''' ≥ '''0'''.
| |
| | |
| We assume that matrix equations of the form
| |
| | |
| : <math> \bold B \bold x = \bold g, \quad (3) </math>
| |
| | |
| where '''g''' is a given column vector, can be solved directly for the vector '''x'''. If (2) represents a regular splitting of '''A''', then the iterative method
| |
| | |
| : <math> \bold B \bold x^{(m+1)} = \bold C \bold x^{(m)} + \bold k, \quad m = 0, 1, 2, \ldots , \quad (4) </math>
| |
| | |
| where '''x'''<sup>(0)</sup> is an arbitrary vector, can be carried out. Equivalently, we write (4) in the form
| |
| | |
| : <math> \bold x^{(m+1)} = \bold B^{-1} \bold C \bold x^{(m)} + \bold B^{-1} \bold k, \quad m = 0, 1, 2, \ldots \quad (5) </math> | |
| | |
| The matrix '''D''' = '''B'''<sup>−1</sup>'''C''' has nonnegative entries if (2) represents a regular splitting of '''A'''.<ref>{{harvtxt|Varga|1960|pp=121–122}}</ref>
| |
| | |
| It can be shown that if '''A'''<sup>−1</sup> > '''0''', then <math>\rho (\bold D)</math> < 1, where <math>\rho (\bold D)</math> represents the [[spectral radius]] of '''D''', and thus '''D''' is a [[convergent matrix]]. As a consequence, the iterative method (5) is necessarily [[Jacobi method#Convergence|convergent]].<ref>{{harvtxt|Varga|1960|pp=122–123}}</ref><ref>{{harvtxt|Varga|1962|p=89}}</ref>
| |
| | |
| If, in addition, the splitting (2) is chosen so that the matrix '''B''' is a [[diagonal matrix]] (with the diagonal entries all non-zero, since '''B''' must be [[Invertible matrix|invertible]]), then '''B''' can be inverted in linear time (see [[Time complexity]]).
| |
| | |
| ==Matrix iterative methods==
| |
| Many iterative methods can be described as a matrix splitting. If the diagonal entries of the matrix '''A''' are all nonzero, and we express the matrix '''A''' as the matrix sum
| |
| | |
| :<math> \bold A = \bold D - \bold U - \bold L, \quad (6) </math>
| |
| | |
| where '''D''' is the diagonal part of '''A''', and '''U''' and '''L''' are respectively strictly upper and lower [[triangular matrix|triangular]] ''n'' × ''n'' matrices, then we have the following.
| |
| | |
| The [[Jacobi method]] can be represented in matrix form as a splitting
| |
| | |
| :<math> \bold x^{(m+1)} = \bold D^{-1}(\bold U + \bold L)\bold x^{(m)} + \bold D^{-1}\bold k. \quad (7) </math><ref>{{harvtxt|Burden|Faires|1993|p=408}}</ref><ref>{{harvtxt|Varga|1962|p=88}}</ref>
| |
| | |
| The [[Gauss-Seidel method]] can be represented in matrix form as a splitting
| |
| :<math> \bold x^{(m+1)} = (\bold D - \bold L)^{-1}\bold U \bold x^{(m)} + (\bold D - \bold L)^{-1}\bold k. \quad (8) </math><ref>{{harvtxt|Burden|Faires|1993|p=411}}</ref><ref>{{harvtxt|Varga|1962|p=88}}</ref> | |
| | |
| The method of [[successive over-relaxation]] can be represented in matrix form as a splitting
| |
| :<math> \bold x^{(m+1)} = (\bold D - \omega \bold L)^{-1}[(1 - \omega) \bold D + \omega \bold U] \bold x^{(m)} + \omega (\bold D - \omega \bold L)^{-1}\bold k. \quad (9) </math><ref>{{harvtxt|Burden|Faires|1993|p=416}}</ref><ref>{{harvtxt|Varga|1962|p=88}}</ref>
| |
| | |
| ==Example==
| |
| | |
| ===Regular splitting===
| |
| In equation (1), let
| |
| :<math>\begin{align}
| |
| & \mathbf{A} = \begin{pmatrix}
| |
| 6 & -2 & -3 \\
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| -1 & 4 & -2 \\
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| -3 & -1 & 5 | |
| \end{pmatrix}, \quad \mathbf{k} = \begin{pmatrix}
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| 5 \\
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| -12 \\
| |
| 10
| |
| \end{pmatrix}. \quad (10)
| |
| \end{align}</math>
| |
| Let us apply the splitting (7) which is used in the Jacobi method: we split '''A''' in such a way that '''B''' consists of ''all'' of the diagonal elements of '''A''', and '''C''' consists of ''all'' of the off-diagonal elements of '''A''', negated. (Of course this is not the only useful way to split a matrix into two matrices.) We have
| |
| :<math>\begin{align}
| |
| & \mathbf{B} = \begin{pmatrix}
| |
| 6 & 0 & 0 \\
| |
| 0 & 4 & 0 \\
| |
| 0 & 0 & 5
| |
| \end{pmatrix}, \quad \mathbf{C} = \begin{pmatrix}
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| 0 & 2 & 3 \\
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| 1 & 0 & 2 \\
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| 3 & 1 & 0
| |
| \end{pmatrix}, \quad (11)
| |
| \end{align}</math>
| |
| :<math>\begin{align}
| |
| & \mathbf{A^{-1}} = \frac{1}{47} \begin{pmatrix}
| |
| 18 & 13 & 16 \\
| |
| 11 & 21 & 15 \\
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| 13 & 12 & 22
| |
| \end{pmatrix}, \quad \mathbf{B^{-1}} = \begin{pmatrix}
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| \frac{1}{6} & 0 & 0 \\[4pt]
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| 0 & \frac{1}{4} & 0 \\[4pt]
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| 0 & 0 & \frac{1}{5}
| |
| \end{pmatrix},
| |
| \end{align}</math>
| |
| :<math>\begin{align} | |
| \mathbf{D} = \mathbf{B^{-1}C} = \begin{pmatrix}
| |
| 0 & \frac{1}{3} & \frac{1}{2} \\[4pt]
| |
| \frac{1}{4} & 0 & \frac{1}{2} \\[4pt]
| |
| \frac{3}{5} & \frac{1}{5} & 0
| |
| \end{pmatrix}, \quad \mathbf{B^{-1}k} = \begin{pmatrix}
| |
| \frac{5}{6} \\[4pt]
| |
| -3 \\[4pt]
| |
| 2
| |
| \end{pmatrix}.
| |
| \end{align}</math>
| |
| Since '''B'''<sup>−1</sup> ≥ '''0''' and '''C''' ≥ '''0''', the splitting (11) is a regular splitting. Since '''A'''<sup>−1</sup> > '''0''', the spectral radius <math>\rho (\bold D)</math> < 1. (The approximate [[eigenvalues]] of '''A''' are <math>\lambda_i</math> ≈ –0.4599820, –0.3397859, 0.7997679.) Hence, the matrix '''D''' is convergent and the method (5) necessarily converges for the problem (10). Note that the diagonal elements of '''A''' are all greater than zero, the off-diagonal elements of '''A''' are all less than zero and '''A''' is [[strictly diagonally dominant]].<ref>{{harvtxt|Burden|Faires|1993|p=371}}</ref>
| |
| | |
| The method (5) applied to the problem (10) then takes the form
| |
| : <math> \bold x^{(m+1)} =
| |
| \begin{align}
| |
| \begin{pmatrix}
| |
| 0 & \frac{1}{3} & \frac{1}{2} \\[4pt]
| |
| \frac{1}{4} & 0 & \frac{1}{2} \\[4pt]
| |
| \frac{3}{5} & \frac{1}{5} & 0
| |
| \end{pmatrix}
| |
| \bold x^{(m)} +
| |
| \begin{pmatrix}
| |
| \frac{5}{6} \\[4pt]
| |
| -3 \\[4pt] | |
| 2
| |
| \end{pmatrix}
| |
| \end{align},
| |
| \quad m = 0, 1, 2, \ldots \quad (12) </math>
| |
| | |
| The exact solution to equation (12) is
| |
| :<math>\begin{align}
| |
| & \mathbf{x} = \begin{pmatrix}
| |
| 2 \\
| |
| -1 \\ | |
| 3
| |
| \end{pmatrix}. \quad (13)
| |
| \end{align}</math>
| |
| The first few iterates for equation (12) are listed in the table below, beginning with '''x'''<sup>(0)</sup> = (0.0, 0.0, 0.0)<sup>T</sup>. From the table one can see that the method is evidently converging to the solution (13), albeit rather slowly.
| |
| | |
| {| class="wikitable" border="1"
| |
| |-
| |
| ! <math>x^{(m)}_1</math>
| |
| ! <math>x^{(m)}_2</math>
| |
| ! <math>x^{(m)}_3</math>
| |
| |-
| |
| | <math>0.0</math>
| |
| | <math>0.0</math>
| |
| | <math>0.0</math>
| |
| |-
| |
| | <math>0.83333</math>
| |
| | <math>-3.0000</math>
| |
| | <math>2.0000</math>
| |
| |-
| |
| | <math>0.83333</math>
| |
| | <math>-1.7917</math>
| |
| | <math>1.9000</math>
| |
| |-
| |
| | <math>1.1861</math>
| |
| | <math>-1.8417</math>
| |
| | <math>2.1417</math>
| |
| |-
| |
| | <math>1.2903</math>
| |
| | <math>-1.6326</math>
| |
| | <math>2.3433</math>
| |
| |-
| |
| | <math>1.4608</math>
| |
| | <math>-1.5058</math>
| |
| | <math>2.4477</math>
| |
| |-
| |
| | <math>1.5553</math>
| |
| | <math>-1.4110</math>
| |
| | <math>2.5753</math>
| |
| |-
| |
| | <math>1.6507</math>
| |
| | <math>-1.3235</math>
| |
| | <math>2.6510</math>
| |
| |-
| |
| | <math>1.7177</math>
| |
| | <math>-1.2618</math>
| |
| | <math>2.7257</math>
| |
| |-
| |
| | <math>1.7756</math>
| |
| | <math>-1.2077</math>
| |
| | <math>2.7783</math>
| |
| |-
| |
| | <math>1.8199</math>
| |
| | <math>-1.1670</math>
| |
| | <math>2.8238</math>
| |
| |}
| |
| | |
| ===Jacobi method===
| |
| As stated above, the Jacobi method (7) is the same as the specific regular splitting (11) demonstrated above.
| |
| | |
| ===Gauss-Seidel method===
| |
| Since the diagonal entries of the matrix '''A''' in problem (10) are all nonzero, we can express the matrix '''A''' as the splitting (6), where
| |
| | |
| :<math>\begin{align}
| |
| & \mathbf{D} = \begin{pmatrix}
| |
| 6 & 0 & 0 \\
| |
| 0 & 4 & 0 \\
| |
| 0 & 0 & 5
| |
| \end{pmatrix}, \quad \mathbf{U} = \begin{pmatrix}
| |
| 0 & 2 & 3 \\
| |
| 0 & 0 & 2 \\
| |
| 0 & 0 & 0
| |
| \end{pmatrix}, \quad \mathbf{L} = \begin{pmatrix}
| |
| 0 & 0 & 0 \\
| |
| 1 & 0 & 0 \\
| |
| 3 & 1 & 0
| |
| \end{pmatrix}. \quad (14)
| |
| \end{align}</math>
| |
| | |
| We then have
| |
| | |
| :<math>\begin{align}
| |
| & \mathbf{(D-L)^{-1}} = \frac{1}{120} \begin{pmatrix}
| |
| 20 & 0 & 0 \\
| |
| 5 & 30 & 0 \\
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| 13 & 6 & 24
| |
| \end{pmatrix},
| |
| \end{align}</math>
| |
| | |
| :<math>\begin{align}
| |
| & \mathbf{(D-L)^{-1}U} = \frac{1}{120} \begin{pmatrix}
| |
| 0 & 40 & 60 \\
| |
| 0 & 10 & 75 \\
| |
| 0 & 26 & 51
| |
| \end{pmatrix}, \quad \mathbf{(D-L)^{-1}k} = \frac{1}{120} \begin{pmatrix}
| |
| 100 \\
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| -335 \\
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| 233
| |
| \end{pmatrix}.
| |
| \end{align}</math>
| |
| | |
| The Gauss-Seidel method (8) applied to the problem (10) takes the form
| |
| | |
| : <math> \bold x^{(m+1)} =
| |
| \begin{align}
| |
| & \frac{1}{120} \begin{pmatrix}
| |
| 0 & 40 & 60 \\
| |
| 0 & 10 & 75 \\
| |
| 0 & 26 & 51
| |
| \end{pmatrix}
| |
| \bold x^{(m)} +
| |
| \frac{1}{120} \begin{pmatrix}
| |
| 100 \\
| |
| -335 \\
| |
| 233
| |
| \end{pmatrix},
| |
| \end{align}
| |
| \quad m = 0, 1, 2, \ldots \quad (15) </math>
| |
| | |
| The first few iterates for equation (15) are listed in the table below, beginning with '''x'''<sup>(0)</sup> = (0.0, 0.0, 0.0)<sup>T</sup>. From the table one can see that the method is evidently converging to the solution (13), somewhat faster than the Jacobi method described above.
| |
| | |
| {| class="wikitable" border="1"
| |
| |-
| |
| ! <math>x^{(m)}_1</math>
| |
| ! <math>x^{(m)}_2</math>
| |
| ! <math>x^{(m)}_3</math>
| |
| |-
| |
| | <math>0.0</math>
| |
| | <math>0.0</math>
| |
| | <math>0.0</math>
| |
| |-
| |
| | <math>0.8333</math>
| |
| | <math>-2.7917</math>
| |
| | <math>1.9417</math>
| |
| |-
| |
| | <math>0.8736</math>
| |
| | <math>-1.8107</math>
| |
| | <math>2.1620</math>
| |
| |-
| |
| | <math>1.3108</math>
| |
| | <math>-1.5913</math>
| |
| | <math>2.4682</math>
| |
| |-
| |
| | <math>1.5370</math>
| |
| | <math>-1.3817</math>
| |
| | <math>2.6459</math>
| |
| |-
| |
| | <math>1.6957</math>
| |
| | <math>-1.2531</math>
| |
| | <math>2.7668</math>
| |
| |-
| |
| | <math>1.7990</math>
| |
| | <math>-1.1668</math>
| |
| | <math>2.8461</math>
| |
| |-
| |
| | <math>1.8675</math>
| |
| | <math>-1.1101</math>
| |
| | <math>2.8985</math>
| |
| |-
| |
| | <math>1.9126</math>
| |
| | <math>-1.0726</math>
| |
| | <math>2.9330</math>
| |
| |-
| |
| | <math>1.9423</math>
| |
| | <math>-1.0479</math>
| |
| | <math>2.9558</math>
| |
| |-
| |
| | <math>1.9619</math>
| |
| | <math>-1.0316</math>
| |
| | <math>2.9708</math>
| |
| |}
| |
| | |
| ===Successive over-relaxation method===
| |
| Let ''ω'' = 1.1. Using the splitting (14) of the matrix '''A''' in problem (10) for the successive over-relaxation method, we have
| |
| | |
| <!-- (D – wL)–1 -->
| |
| :<math>\begin{align}
| |
| & \mathbf{(D-\omega L)^{-1}} = \frac{1}{12} \begin{pmatrix}
| |
| 2 & 0 & 0 \\
| |
| 0.55 & 3 & 0 \\
| |
| 1.441 & 0.66 & 2.4
| |
| \end{pmatrix},
| |
| \end{align}</math>
| |
| | |
| <!-- (D – wL)–1[(1 – w)D + wU] -->
| |
| :<math>\begin{align} | |
| & \mathbf{(D-\omega L)^{-1}[(1-\omega )D+\omega U]} = \frac{1}{12} \begin{pmatrix}
| |
| -1.2 & 4.4 & 6.6 \\
| |
| -0.33 & 0.01 & 8.415 \\
| |
| -0.8646 & 2.9062 & 5.0073
| |
| \end{pmatrix},
| |
| \end{align}</math>
| |
| | |
| <!-- w(D – wL)–1k -->
| |
| :<math>\begin{align}
| |
| & \mathbf{\omega (D-\omega L)^{-1}k} = \frac{1}{12} \begin{pmatrix}
| |
| 11 \\
| |
| -36.575 \\
| |
| 25.6135
| |
| \end{pmatrix}.
| |
| \end{align}</math>
| |
| | |
| The successive over-relaxation method (9) applied to the problem (10) takes the form
| |
| | |
| : <math> \bold x^{(m+1)} =
| |
| \begin{align}
| |
| & \frac{1}{12} \begin{pmatrix}
| |
| -1.2 & 4.4 & 6.6 \\
| |
| -0.33 & 0.01 & 8.415 \\
| |
| -0.8646 & 2.9062 & 5.0073
| |
| \end{pmatrix}
| |
| \bold x^{(m)} +
| |
| \frac{1}{12} \begin{pmatrix}
| |
| 11 \\
| |
| -36.575 \\
| |
| 25.6135
| |
| \end{pmatrix},
| |
| \end{align}
| |
| \quad m = 0, 1, 2, \ldots \quad (16) </math>
| |
| | |
| The first few iterates for equation (16) are listed in the table below, beginning with '''x'''<sup>(0)</sup> = (0.0, 0.0, 0.0)<sup>T</sup>. From the table one can see that the method is evidently converging to the solution (13), slightly faster than the Gauss-Seidel method described above.
| |
| | |
| {| class="wikitable" border="1"
| |
| |-
| |
| ! <math>x^{(m)}_1</math>
| |
| ! <math>x^{(m)}_2</math>
| |
| ! <math>x^{(m)}_3</math>
| |
| |-
| |
| | <math>0.0</math>
| |
| | <math>0.0</math>
| |
| | <math>0.0</math>
| |
| |-
| |
| | <math>0.9167</math>
| |
| | <math>-3.0479</math>
| |
| | <math>2.1345</math>
| |
| |-
| |
| | <math>0.8814</math>
| |
| | <math>-1.5788</math>
| |
| | <math>2.2209</math>
| |
| |-
| |
| | <math>1.4711</math>
| |
| | <math>-1.5161</math>
| |
| | <math>2.6153</math>
| |
| |-
| |
| | <math>1.6521</math>
| |
| | <math>-1.2557</math>
| |
| | <math>2.7526</math>
| |
| |-
| |
| | <math>1.8050</math>
| |
| | <math>-1.1641</math>
| |
| | <math>2.8599</math>
| |
| |-
| |
| | <math>1.8823</math>
| |
| | <math>-1.0930</math>
| |
| | <math>2.9158</math>
| |
| |-
| |
| | <math>1.9314</math>
| |
| | <math>-1.0559</math>
| |
| | <math>2.9508</math>
| |
| |-
| |
| | <math>1.9593</math>
| |
| | <math>-1.0327</math>
| |
| | <math>2.9709</math>
| |
| |-
| |
| | <math>1.9761</math>
| |
| | <math>-1.0185</math>
| |
| | <math>2.9829</math>
| |
| |-
| |
| | <math>1.9862</math>
| |
| | <math>-1.0113</math>
| |
| | <math>2.9901</math>
| |
| |}
| |
| | |
| ==See also==
| |
| *[[Matrix decomposition]]
| |
| *[[M-matrix]]
| |
| *[[Stieltjes matrix]]
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| * {{citation | first1 = Richard L. | last1 = Burden | first2 = J. Douglas | last2 = Faires | year = 1993 | isbn = 0-534-93219-3 | title = Numerical Analysis | edition = 5th | publisher = [[Prindle, Weber and Schmidt]] | location = Boston }}.
| |
| | |
| * {{Cite book | first1 = Richard S. | last1 = Varga | chapter = Factorization and Normalized Iterative Methods | title = Boundary Problems in Differential Equations | editor1-last = Langer | editor1-first = Rudolph E. | publisher = [[University of Wisconsin Press]] | location = Madison | pages = 121–142 | year = 1960 | lccn = 60-60003}}
| |
| | |
| * {{citation | first1 = Richard S. | last1 = Varga | title = Matrix Iterative Analysis | publisher = [[Prentice-Hall]] | location = New Jersey | year = 1962 | lccn = 62-21277}}.
| |
| | |
| {{Numerical linear algebra}}
| |
| | |
| [[Category:Matrices]]
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| [[Category:Numerical linear algebra]]
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| [[Category:Relaxation (iterative methods)]]
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