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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 (e.g., 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.

Regular splittings

We seek to solve the matrix equation

$\mathbf{A}\mathbf{x}=\mathbf{k},(1)$

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

$\mathbf{A}=\mathbf{B}-\mathbf{C},(2)$

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 and only if B⁻¹ ≥ 0 and C ≥ 0.

We assume that matrix equations of the form

$\mathbf{B}\mathbf{x}=\mathbf{g},(3)$

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

$\mathbf{B}{\mathbf{x}}^{(m+1)}=\mathbf{C}{\mathbf{x}}^{(m)}+\mathbf{k},m=0,1,2,\dots ,(4)$

where x⁽⁰⁾ is an arbitrary vector, can be carried out. Equivalently, we write (4) in the form

${\mathbf{x}}^{(m+1)}={\mathbf{B}}^{-1}\mathbf{C}{\mathbf{x}}^{(m)}+{\mathbf{B}}^{-1}\mathbf{k},m=0,1,2,\dots (5)$

The matrix D = B⁻¹C has nonnegative entries if (2) represents a regular splitting of A.

It can be shown that if A⁻¹ > 0, then $\rho (\mathbf{D})$ < 1, where $\rho (\mathbf{D})$ represents the spectral radius of D, and thus D is a convergent matrix. As a consequence, the iterative method (5) is necessarily convergent.

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), 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

$\mathbf{A}=\mathbf{D}-\mathbf{U}-\mathbf{L},(6)$

where D is the diagonal part of A, and U and L are respectively strictly upper and lower triangular n × n matrices, then we have the following.

The Jacobi method can be represented in matrix form as a splitting

${\mathbf{x}}^{(m+1)}={\mathbf{D}}^{-1}(\mathbf{U}+\mathbf{L}){\mathbf{x}}^{(m)}+{\mathbf{D}}^{-1}\mathbf{k}.(7)$

The Gauss-Seidel method can be represented in matrix form as a splitting

${\mathbf{x}}^{(m+1)}=(\mathbf{D}-\mathbf{L}{)}^{-1}\mathbf{U}{\mathbf{x}}^{(m)}+(\mathbf{D}-\mathbf{L}{)}^{-1}\mathbf{k}.(8)$

The method of successive over-relaxation can be represented in matrix form as a splitting

${\mathbf{x}}^{(m+1)}=(\mathbf{D}-\omega \mathbf{L}{)}^{-1}[(1-\omega )\mathbf{D}+\omega \mathbf{U}]{\mathbf{x}}^{(m)}+\omega (\mathbf{D}-\omega \mathbf{L}{)}^{-1}\mathbf{k}.(9)$

Example

Regular splitting

In equation (1), let

$\begin{array}{rl}& \mathbf{A}=\left(\begin{array}{ccc}6& -2& -3\\ -1& 4& -2\\ -3& -1& 5\end{array}\right),\mathbf{k}=\left(\begin{array}{c}5\\ -12\\ 10\end{array}\right).(10)\end{array}$

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

$\begin{array}{rl}& \mathbf{B}=\left(\begin{array}{ccc}6& 0& 0\\ 0& 4& 0\\ 0& 0& 5\end{array}\right),\mathbf{C}=\left(\begin{array}{ccc}0& 2& 3\\ 1& 0& 2\\ 3& 1& 0\end{array}\right),(11)\end{array}$

$\begin{array}{rl}& {\mathbf{A}}^{\mathbf{-}\mathbf{1}}=\frac{1}{47}\left(\begin{array}{ccc}18& 13& 16\\ 11& 21& 15\\ 13& 12& 22\end{array}\right),{\mathbf{B}}^{\mathbf{-}\mathbf{1}}=\left(\begin{array}{ccc}\frac{1}{6}& 0& 0\\ 0& \frac{1}{4}& 0\\ 0& 0& \frac{1}{5}\end{array}\right),\end{array}$

$\begin{array}{r}\mathbf{D}={\mathbf{B}}^{\mathbf{-}\mathbf{1}}\mathbf{C}=\left(\begin{array}{ccc}0& \frac{1}{3}& \frac{1}{2}\\ \frac{1}{4}& 0& \frac{1}{2}\\ \frac{3}{5}& \frac{1}{5}& 0\end{array}\right),{\mathbf{B}}^{\mathbf{-}\mathbf{1}}\mathbf{k}=\left(\begin{array}{c}\frac{5}{6}\\ -3\\ 2\end{array}\right).\end{array}$

Since B⁻¹ ≥ 0 and C ≥ 0, the splitting (11) is a regular splitting. Since A⁻¹ > 0, the spectral radius $\rho (\mathbf{D})$ < 1. (The approximate eigenvalues of A are ${\lambda }_i$ ≈ –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.

The method (5) applied to the problem (10) then takes the form

${\mathbf{x}}^{(m+1)}=\begin{array}{r}\left(\begin{array}{ccc}0& \frac{1}{3}& \frac{1}{2}\\ \frac{1}{4}& 0& \frac{1}{2}\\ \frac{3}{5}& \frac{1}{5}& 0\end{array}\right){\mathbf{x}}^{(m)}+\left(\begin{array}{c}\frac{5}{6}\\ -3\\ 2\end{array}\right)\end{array},m=0,1,2,\dots (12)$

The exact solution to equation (12) is

$\begin{array}{rl}& \mathbf{x}=\left(\begin{array}{c}2\\ -1\\ 3\end{array}\right).(13)\end{array}$

The first few iterates for equation (12) are listed in the table below, beginning with x⁽⁰⁾ = (0.0, 0.0, 0.0)^T. From the table one can see that the method is evidently converging to the solution (13), albeit rather slowly.

$x_1^{(m)}$	$x_2^{(m)}$	$x_3^{(m)}$
$0.0$	$0.0$	$0.0$
$0.83333$	$-3.0000$	$2.0000$
$0.83333$	$-1.7917$	$1.9000$
$1.1861$	$-1.8417$	$2.1417$
$1.2903$	$-1.6326$	$2.3433$
$1.4608$	$-1.5058$	$2.4477$
$1.5553$	$-1.4110$	$2.5753$
$1.6507$	$-1.3235$	$2.6510$
$1.7177$	$-1.2618$	$2.7257$
$1.7756$	$-1.2077$	$2.7783$
$1.8199$	$-1.1670$	$2.8238$

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

$\begin{array}{rl}& \mathbf{D}=\left(\begin{array}{ccc}6& 0& 0\\ 0& 4& 0\\ 0& 0& 5\end{array}\right),\mathbf{U}=\left(\begin{array}{ccc}0& 2& 3\\ 0& 0& 2\\ 0& 0& 0\end{array}\right),\mathbf{L}=\left(\begin{array}{ccc}0& 0& 0\\ 1& 0& 0\\ 3& 1& 0\end{array}\right).(14)\end{array}$

We then have

$\begin{array}{rl}& \mathbf{(}\mathbf{D}\mathbf{-}\mathbf{L}{\mathbf{)}}^{\mathbf{-}\mathbf{1}}=\frac{1}{120}\left(\begin{array}{ccc}20& 0& 0\\ 5& 30& 0\\ 13& 6& 24\end{array}\right),\end{array}$

$\begin{array}{rl}& \mathbf{(}\mathbf{D}\mathbf{-}\mathbf{L}{\mathbf{)}}^{\mathbf{-}\mathbf{1}}\mathbf{U}=\frac{1}{120}\left(\begin{array}{ccc}0& 40& 60\\ 0& 10& 75\\ 0& 26& 51\end{array}\right),\mathbf{(}\mathbf{D}\mathbf{-}\mathbf{L}{\mathbf{)}}^{\mathbf{-}\mathbf{1}}\mathbf{k}=\frac{1}{120}\left(\begin{array}{c}100\\ -335\\ 233\end{array}\right).\end{array}$

The Gauss-Seidel method (8) applied to the problem (10) takes the form

${\mathbf{x}}^{(m+1)}=\begin{array}{rl}& \frac{1}{120}\left(\begin{array}{ccc}0& 40& 60\\ 0& 10& 75\\ 0& 26& 51\end{array}\right){\mathbf{x}}^{(m)}+\frac{1}{120}\left(\begin{array}{c}100\\ -335\\ 233\end{array}\right),\end{array}m=0,1,2,\dots (15)$

The first few iterates for equation (15) are listed in the table below, beginning with x⁽⁰⁾ = (0.0, 0.0, 0.0)^T. From the table one can see that the method is evidently converging to the solution (13), somewhat faster than the Jacobi method described above.

$x_1^{(m)}$	$x_2^{(m)}$	$x_3^{(m)}$
$0.0$	$0.0$	$0.0$
$0.8333$	$-2.7917$	$1.9417$
$0.8736$	$-1.8107$	$2.1620$
$1.3108$	$-1.5913$	$2.4682$
$1.5370$	$-1.3817$	$2.6459$
$1.6957$	$-1.2531$	$2.7668$
$1.7990$	$-1.1668$	$2.8461$
$1.8675$	$-1.1101$	$2.8985$
$1.9126$	$-1.0726$	$2.9330$
$1.9423$	$-1.0479$	$2.9558$
$1.9619$	$-1.0316$	$2.9708$

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

$\begin{array}{rl}& \mathbf{(}\mathbf{D}\mathbf{-}\mathit{\omega }\mathbf{L}{\mathbf{)}}^{\mathbf{-}\mathbf{1}}=\frac{1}{12}\left(\begin{array}{ccc}2& 0& 0\\ 0.55& 3& 0\\ 1.441& 0.66& 2.4\end{array}\right),\end{array}$

$\begin{array}{rl}& \mathbf{(}\mathbf{D}\mathbf{-}\mathit{\omega }\mathbf{L}{\mathbf{)}}^{\mathbf{-}\mathbf{1}}\mathbf{[}\mathbf{(}\mathbf{1}\mathbf{-}\mathit{\omega }\mathbf{)}\mathbf{D}+\mathit{\omega }\mathbf{U}\mathbf{]}=\frac{1}{12}\left(\begin{array}{ccc}-1.2& 4.4& 6.6\\ -0.33& 0.01& 8.415\\ -0.8646& 2.9062& 5.0073\end{array}\right),\end{array}$

$\begin{array}{rl}& \mathit{\omega }\mathbf{(}\mathbf{D}\mathbf{-}\mathit{\omega }\mathbf{L}{\mathbf{)}}^{\mathbf{-}\mathbf{1}}\mathbf{k}=\frac{1}{12}\left(\begin{array}{c}11\\ -36.575\\ 25.6135\end{array}\right).\end{array}$

The successive over-relaxation method (9) applied to the problem (10) takes the form

${\mathbf{x}}^{(m+1)}=\begin{array}{rl}& \frac{1}{12}\left(\begin{array}{ccc}-1.2& 4.4& 6.6\\ -0.33& 0.01& 8.415\\ -0.8646& 2.9062& 5.0073\end{array}\right){\mathbf{x}}^{(m)}+\frac{1}{12}\left(\begin{array}{c}11\\ -36.575\\ 25.6135\end{array}\right),\end{array}m=0,1,2,\dots (16)$

The first few iterates for equation (16) are listed in the table below, beginning with x⁽⁰⁾ = (0.0, 0.0, 0.0)^T. 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.

$x_1^{(m)}$	$x_2^{(m)}$	$x_3^{(m)}$
$0.0$	$0.0$	$0.0$
$0.9167$	$-3.0479$	$2.1345$
$0.8814$	$-1.5788$	$2.2209$
$1.4711$	$-1.5161$	$2.6153$
$1.6521$	$-1.2557$	$2.7526$
$1.8050$	$-1.1641$	$2.8599$
$1.8823$	$-1.0930$	$2.9158$
$1.9314$	$-1.0559$	$2.9508$
$1.9593$	$-1.0327$	$2.9709$
$1.9761$	$-1.0185$	$2.9829$
$1.9862$	$-1.0113$	$2.9901$

See also

• Matrix decomposition
• M-matrix
• Stieltjes matrix

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.