逐次超松弛迭代法

数值线性代数中，逐次超松弛（successive over-relaxation，SOR）迭代法是高斯-赛德尔迭代的一种变体，用于求解线性方程组。类似方法也可用于任何缓慢收敛的迭代过程。

SOR迭代法由David M. Young Jr.和Stanley P. Frankel在1950年同时独立提出，目的是在计算机上自动求解线性方程组。之前，人们已经为计算员的计算开发过超松弛法，如路易斯·弗莱·理查德森的方法以及R. V. Southwell开发的方法。但这些方法需要一定专业知识确保求解的收敛，不适用于计算机编程。David M. Young Jr.的论文对这些方面进行了探讨。^[1]

形式化

给定n个线性方程组成的方系统：

A\mathbf {x} =\mathbf {b}

其中

A={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\cdots &a_{nn}\end{bmatrix}},\qquad \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}},\qquad \mathbf {b} ={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{n}\end{bmatrix}}.

则A可分解为对角矩阵D、严格上下三角矩阵U、L：

A=D+L+U,

其中

D={\begin{bmatrix}a_{11}&0&\cdots &0\\0&a_{22}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &a_{nn}\end{bmatrix}},\quad L={\begin{bmatrix}0&0&\cdots &0\\a_{21}&0&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\cdots &0\end{bmatrix}},\quad U={\begin{bmatrix}0&a_{12}&\cdots &a_{1n}\\0&0&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &0\end{bmatrix}}.

线性方程组可重写为

(D+\omega L)\mathbf {x} =\omega \mathbf {b} -[\omega U+(\omega -1)D]\mathbf {x}

其中 $\omega >1$ 是常数，称作松弛因子（relaxation factor）。

逐次超松弛迭代法可以通过迭代逼近x的精确解，可分析地写作

\mathbf {x} ^{(k+1)}=(D+\omega L)^{-1}{\big (}\omega \mathbf {b} -[\omega U+(\omega -1)D]\mathbf {x} ^{(k)}{\big )}=L_{\omega }\mathbf {x} ^{(k)}+\mathbf {c} ,

其中 $\mathbf {x} ^{(k)}$ 是 $\mathbf {x}$ 的第k次迭代值， $\mathbf {x} ^{(k+1)}$ 是 $\mathbf {x}$ 下一次迭代所得的值。利用 $(D+\omega L)$ 的三角形，可用向前替换法依次计算 $\mathbf {x} ^{(k+1)}$ 的元素：

x_{i}^{(k+1)}=(1-\omega )x_{i}^{(k)}+{\frac {\omega }{a_{ii}}}\left(b_{i}-\sum _{j<i}a_{ij}x_{j}^{(k+1)}-\sum _{j>i}a_{ij}x_{j}^{(k)}\right),\quad i=1,2,\ldots ,n.

收敛性

松弛因子 $\omega$ 的选择并不容易，取决于系数矩阵的性质。1947年，亚历山大·马雅科维奇·奥斯特洛夫斯基证明，若A是对称正定矩阵，则 $\forall 0<\omega <2,\ \rho (L_{\omega })<1$ 。因此，迭代过程将收敛，但更高的收敛速度更有价值。

收敛速度

SOR法的收敛速度可通过分析得出。假设^[2]^[3]

松弛因子适当： $\omega \in (0,2)$
雅可比法迭代矩阵 $C_{\text{Jac}}:=I-D^{-1}A$ 只有实特征值
雅可比法收敛： $\mu :=\rho (C_{\text{Jac}})<1$
矩阵分解 $A=D+L+U$ 满足 $\forall z\in \mathbb {C} \setminus \{0\},\ \lambda \in \mathbb {C} ,\ \operatorname {det} (\lambda D+zL+{\tfrac {1}{z}}U)=\operatorname {det} (\lambda D+L+U)$

则收敛速度可表为

\rho (C_{\omega })={\begin{cases}{\frac {1}{4}}\left(\omega \mu +{\sqrt {\omega ^{2}\mu ^{2}-4(\omega -1)}}\right)^{2}\,,&0<\omega \leq \omega _{\text{opt}}\\\omega -1\,,&\omega _{\text{opt}}<\omega <2\end{cases}}

最优松弛因子是

\omega _{\text{opt}}:=1+\left({\frac {\mu }{1+{\sqrt {1-\mu ^{2}}}}}\right)^{2}=1+{\frac {\mu ^{2}}{4}}+O(\mu ^{3})\,.

特别地， $\omega =1$ 时SOR法即退化为高斯-赛德尔迭代，有 $\rho (C_{\omega })=\mu ^{2}=\rho (C_{\text{Jac}})^{2}$ 。对最优的 $\omega$ ，有 $\rho (C_{\omega })={\frac {1-{\sqrt {1-\mu ^{2}}}}{1+{\sqrt {1-\mu ^{2}}}}}={\frac {\mu ^{2}}{4}}+O(\mu ^{3})$ ，表明SOR法的效率约是高斯-赛德尔迭代的4倍。

最后一条假设对三对角矩阵也满足，因为 $Z(\lambda D+L+U)Z^{-1}=\lambda D+zL+{\tfrac {1}{z}}U$ 对对角阵Z，其元素 $Z_{ii}=z^{i-1}$ 、 $\operatorname {det} (\lambda D+L+U)=\operatorname {det} (Z(\lambda D+L+U)Z^{-1})$ 。

算法

由于此算法中，元素可在迭代过程中被覆盖，所以只需一个存储向量，不需要向量索引。

输入： $A$ ,  $b$ ,  $ω$ 
输出： $φ$ 

选择初始解 $φ$ 
repeat until convergence
    for  $i$  from 1 until  $n$  do
        set  $σ$  to 0
        for  $j$  from 1 until  $n$  do
            if  $j$  ≠  $i$  then
                set  $σ$  to  $σ + a ij φ j$ 
            end if
        end ( $j$ -loop)
        set  $φ i$  to  $(1 - ω) φ i + ω (b i - σ) / a ii$ 
    end ( $i$ -loop)
    check if convergence is reached
end (repeat)

注意： $(1-\omega )\phi _{i}+{\frac {\omega }{a_{ii}}}(b_{i}-\sigma )$ 也可写作 $\phi _{i}+\omega \left({\frac {b_{i}-\sigma }{a_{ii}}}-\phi _{i}\right)$ ，这样每次外层for循环可以省去一次乘法。

例子

解线性方程组

{\begin{aligned}4x_{1}-x_{2}-6x_{3}+0x_{4}&=2,\\-5x_{1}-4x_{2}+10x_{3}+8x_{4}&=21,\\0x_{1}+9x_{2}+4x_{3}-2x_{4}&=-12,\\1x_{1}+0x_{2}-7x_{3}+5x_{4}&=-6.\end{aligned}}

择松弛因子 $\omega =0.5$ 与初始解 $\phi =(0,0,0,0)$ 。由SOR算法可得下表，在38步取得精确解(3, −2, 2, 1)。

迭代	$x_{1}$	$x_{2}$	$x_{3}$	$x_{4}$
01	0.25	−2.78125	1.6289062	0.5152344
02	1.2490234	−2.2448974	1.9687712	0.9108547
03	2.070478	−1.6696789	1.5904881	0.76172125
...	...	...	...	...
37	2.9999998	−2.0	2.0	1.0
38	3.0	−2.0	2.0	1.0

用Common Lisp的简单实现：

;; 默认浮点格式设为long-float，以确保在更大范围数字上正确运行
(setf *read-default-float-format* 'long-float)

(defparameter +MAXIMUM-NUMBER-OF-ITERATIONS+ 100
  "The number of iterations beyond which the algorithm should cease its
   operation, regardless of its current solution. A higher number of
   iterations might provide a more accurate result, but imposes higher
   performance requirements.")

(declaim (type (integer 0 *) +MAXIMUM-NUMBER-OF-ITERATIONS+))

(defun get-errors (computed-solution exact-solution)
  "For each component of the COMPUTED-SOLUTION vector, retrieves its
   error with respect to the expected EXACT-SOLUTION vector, returning a
   vector of error values.
   ---
   While both input vectors should be equal in size, this condition is
   not checked and the shortest of the twain determines the output
   vector's number of elements.
   ---
   The established formula is the following:
     Let resultVectorSize = min(computedSolution.length, exactSolution.length)
     Let resultVector     = new vector of resultVectorSize
     For i from 0 to (resultVectorSize - 1)
       resultVector[i] = exactSolution[i] - computedSolution[i]
     Return resultVector"
  (declare (type (vector number *) computed-solution))
  (declare (type (vector number *) exact-solution))
  (map '(vector number *) #'- exact-solution computed-solution))

(defun is-convergent (errors &key (error-tolerance 0.001))
  "Checks whether the convergence is reached with respect to the
   ERRORS vector which registers the discrepancy betwixt the computed
   and the exact solution vector.
   ---
   The convergence is fulfilled if and only if each absolute error
   component is less than or equal to the ERROR-TOLERANCE, that is:
   For all e in ERRORS, it holds: abs(e) <= errorTolerance."
  (declare (type (vector number *) errors))
  (declare (type number            error-tolerance))
  (flet ((error-is-acceptable (error)
          (declare (type number error))
          (<= (abs error) error-tolerance)))
    (every #'error-is-acceptable errors)))

(defun make-zero-vector (size)
  "Creates and returns a vector of the SIZE with all elements set to 0."
  (declare (type (integer 0 *) size))
  (make-array size :initial-element 0.0 :element-type 'number))

(defun successive-over-relaxation (A b omega
                                   &key (phi (make-zero-vector (length b)))
                                        (convergence-check
                                          #'(lambda (iteration phi)
                                              (declare (ignore phi))
                                              (>= iteration +MAXIMUM-NUMBER-OF-ITERATIONS+))))
  "Implements the successive over-relaxation (SOR) method, applied upon
   the linear equations defined by the matrix A and the right-hand side
   vector B, employing the relaxation factor OMEGA, returning the
   calculated solution vector.
   ---
   The first algorithm step, the choice of an initial guess PHI, is
   represented by the optional keyword parameter PHI, which defaults
   to a zero-vector of the same structure as B. If supplied, this
   vector will be destructively modified. In any case, the PHI vector
   constitutes the function's result value.
   ---
   The terminating condition is implemented by the CONVERGENCE-CHECK,
   an optional predicate
     lambda(iteration phi) => generalized-boolean
   which returns T, signifying the immediate termination, upon achieving
   convergence, or NIL, signaling continuant operation, otherwise. In
   its default configuration, the CONVERGENCE-CHECK simply abides the
   iteration's ascension to the ``+MAXIMUM-NUMBER-OF-ITERATIONS+'',
   ignoring the achieved accuracy of the vector PHI."
  (declare (type (array  number (* *)) A))
  (declare (type (vector number *)     b))
  (declare (type number                omega))
  (declare (type (vector number *)     phi))
  (declare (type (function ((integer 1 *)
                            (vector number *))
                           *)
                 convergence-check))
  (let ((n (array-dimension A 0)))
    (declare (type (integer 0 *) n))
    (loop for iteration from 1 by 1 do
      (loop for i from 0 below n by 1 do
        (let ((rho 0))
          (declare (type number rho))
          (loop for j from 0 below n by 1 do
            (when (/= j i)
              (let ((a[ij]  (aref A i j))
                    (phi[j] (aref phi j)))
                (incf rho (* a[ij] phi[j])))))
          (setf (aref phi i)
                (+ (* (- 1 omega)
                      (aref phi i))
                   (* (/ omega (aref A i i))
                      (- (aref b i) rho))))))
      (format T "~&~d. solution = ~a" iteration phi)
      ;; Check if convergence is reached.
      (when (funcall convergence-check iteration phi)
        (return))))
  (the (vector number *) phi))

;; Summon the function with the exemplary parameters.
(let ((A              (make-array (list 4 4)
                        :initial-contents
                        '((  4  -1  -6   0 )
                          ( -5  -4  10   8 )
                          (  0   9   4  -2 )
                          (  1   0  -7   5 ))))
      (b              (vector 2 21 -12 -6))
      (omega          0.5)
      (exact-solution (vector 3 -2 2 1)))
  (successive-over-relaxation
    A b omega
    :convergence-check
    #'(lambda (iteration phi)
        (declare (type (integer 0 *)     iteration))
        (declare (type (vector number *) phi))
        (let ((errors (get-errors phi exact-solution)))
          (declare (type (vector number *) errors))
          (format T "~&~d. errors   = ~a" iteration errors)
          (or (is-convergent errors :error-tolerance 0.0)
              (>= iteration +MAXIMUM-NUMBER-OF-ITERATIONS+))))))

上述伪代码的简单Python实现。

import numpy as np
from scipy import linalg

def sor_solver(A, b, omega, initial_guess, convergence_criteria):
    """
    This is an implementation of the pseudo-code provided in the Wikipedia article.
    Arguments:
        A: nxn numpy matrix.
        b: n dimensional numpy vector.
        omega: relaxation factor.
        initial_guess: An initial solution guess for the solver to start with.
        convergence_criteria: The maximum discrepancy acceptable to regard the current solution as fitting.
    Returns:
        phi: solution vector of dimension n.
    """
    step = 0
    phi = initial_guess[:]
    residual = linalg.norm(A @ phi - b)  # Initial residual
    while residual > convergence_criteria:
        for i in range(A.shape[0]):
            sigma = 0
            for j in range(A.shape[1]):
                if j != i:
                    sigma += A[i, j] * phi[j]
            phi[i] = (1 - omega) * phi[i] + (omega / A[i, i]) * (b[i] - sigma)
        residual = linalg.norm(A @ phi - b)
        step += 1
        print("Step {} Residual: {:10.6g}".format(step, residual))
    return phi

# An example case that mirrors the one in the Wikipedia article
residual_convergence = 1e-8
omega = 0.5  # Relaxation factor

A = np.array([[4, -1, -6, 0],
              [-5, -4, 10, 8],
              [0, 9, 4, -2],
              [1, 0, -7, 5]])

b = np.array([2, 21, -12, -6])

initial_guess = np.zeros(4)

phi = sor_solver(A, b, omega, initial_guess, residual_convergence)
print(phi)

对称逐次超松弛

对对称矩阵A，其中

U=L^{T},\,

有对称逐次超松弛迭代法（SSOR）：

P=\left({\frac {D}{\omega }}+L\right){\frac {\omega }{2-\omega }}D^{-1}\left({\frac {D}{\omega }}+U\right),

迭代法为

\mathbf {x} ^{k+1}=\mathbf {x} ^{k}-\gamma ^{k}P^{-1}(A\mathbf {x} ^{k}-\mathbf {b} ),\ k\geq 0.

SOR与SSOR法都来自David M. Young Jr.

其他应用

任何迭代法都可应用相似技巧。若原迭代的形式为

x_{n+1}=f(x_{n})

则可将其改为

x_{n+1}^{\mathrm {SOR} }=(1-\omega )x_{n}^{\mathrm {SOR} }+\omega f(x_{n}^{\mathrm {SOR} }).

但若将x视作完整向量，则上述解线性方程组的公式不是这种公式的特例。此公式基础上，计算下一个向量的公式是

\mathbf {x} ^{(k+1)}=(1-\omega )\mathbf {x} ^{(k)}+\omega L_{*}^{-1}(\mathbf {b} -U\mathbf {x} ^{(k)}),

其中 $L_{*}=L+D$ 。 $\omega >1$ 用于加快收敛速度， $\omega <1$ 可使发散的迭代收敛或加快过调（overshoot）过程的收敛。有多种方法可根据观察到的收敛过程行为，自适应地调整松弛因子 $\omega$ 。这些方法通常只对一部分问题有效。

另见

注释

^ Young, David M., Iterative methods for solving partial difference equations of elliptical type (PDF), PhD thesis, Harvard University, 1950-05-01 [2009-06-15]
^ Hackbusch, Wolfgang. 4.6.2. Iterative Solution of Large Sparse Systems of Equations | SpringerLink. Applied Mathematical Sciences 95. 2016. ISBN 978-3-319-28481-1. doi:10.1007/978-3-319-28483-5 （英国英语）.
^ Greenbaum, Anne. 10.1. Iterative Methods for Solving Linear Systems. Frontiers in Applied Mathematics 17. 1997. ISBN 978-0-89871-396-1. doi:10.1137/1.9781611970937 （英国英语）.

参考文献

Abraham Berman, Robert J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, 1994, SIAM. ISBN 0-89871-321-8.
Black, Noel. Successive Overrelaxation Method. MathWorld.
A. Hadjidimos, Successive overrelaxation (SOR) and related methods, Journal of Computational and Applied Mathematics 123 (2000), 177–199.
Yousef Saad, Iterative Methods for Sparse Linear Systems, 1st edition, PWS, 1996.
Netlib's copy of "Templates for the Solution of Linear Systems", by Barrett et al.
Richard S. Varga 2002 Matrix Iterative Analysis, Second ed. (of 1962 Prentice Hall edition), Springer-Verlag.
David M. Young Jr. Iterative Solution of Large Linear Systems, Academic Press, 1971. (reprinted by Dover, 2003)

外部链接

Module for the SOR Method
Tridiagonal linear system solver based on SOR, in C++

[1] Young, David M., Iterative methods for solving partial difference equations of elliptical type (PDF), PhD thesis, Harvard University, 1950-05-01 [2009-06-15]

[2] Hackbusch, Wolfgang. 4.6.2. Iterative Solution of Large Sparse Systems of Equations | SpringerLink. Applied Mathematical Sciences 95. 2016. ISBN 978-3-319-28481-1. doi:10.1007/978-3-319-28483-5 （英国英语）.

[3] Greenbaum, Anne. 10.1. Iterative Methods for Solving Linear Systems. Frontiers in Applied Mathematics 17. 1997. ISBN 978-0-89871-396-1. doi:10.1137/1.9781611970937 （英国英语）.

[1]

[2]

[3]