successive over relaxation


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by Yonaba

in lua

Source Code

-- Successive Over Relaxation algorithm implementation
-- See : http://en.wikipedia.org/wiki/Successive_over-relaxation

-- Creates a vector of values
local function vector(len, v)
  local x = {}
  for i = 1, len do x[i] = v end
  return x
end

-- Computes the norm of a given vector
local function norm(v)
  local n = 0
  for i, _v in ipairs(v) do
    n = n + (_v * _v)
  end
  return math.sqrt(n)
end

-- Solves the given matrix using SOR algorithm
-- m       : a matrix A in [A]*[x] = [b]
-- b       : the result vector in [A]*[x] = [b]
-- w       : the relaxation parameter, defaults to 1.86
-- eps     : the convergence criterion, defaults to 1e-6
-- maxiter : the maximum number of iterations, defaults to 1e4
-- return  : a solution vector
local function SOR(m, b, w, eps, maxiter)
  local n = #m
  local x = vector(n, 0)
  local q, p, sum
  local t = 0

  w = w or 1.86
  eps = eps or 1e-6
  maxiter = maxiter or 1e4

  repeat
    t = t + 1
    q = norm(x)
    for i =1, n do
      sum = 0
      for j = 1, n do
        if (i ~= j) then
          sum = sum + m[i][j] * x[j]
        end
      end
      x[i] = (1 - w) * x[i] + (w / m[i][i]) * (b[i] - sum)
    end
    p = norm(x)
  until (math.abs(p - q) < eps) or (t >= maxiter)

  return x
end

return SOR
-- Tests for sor.lua
local sor = require 'sor'

local total, pass = 0, 0

local function dec(str, len)
  return #str < len
     and str .. (('.'):rep(len-#str))
      or str:sub(1,len)
end

local function run(message, f)
  total = total + 1
  local ok, err = pcall(f)
  if ok then pass = pass + 1 end
  local status = ok and 'PASSED' or 'FAILED'
  print(('%02d. %68s: %s'):format(total, dec(message,68), status))
end

-- Fuzzy equality test
local function fuzzy_eq(a, b, eps)
  eps = eps or 1e-6
  return math.abs(a - b) < eps
end

run('Testing SOR', function()
  local m = {
    {-4,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0},
    {1,-4,1,0,0,1,0,0,0,0,0,0,0,0,0,0},
    {0,1,-4,1,0,0,1,0,0,0,0,0,0,0,0,0},
    {0,0,1,-4,0,0,0,1,0,0,0,0,0,0,0,0},
    {1,0,0,0,-4,1,0,0,1,0,0,0,0,0,0,0},
    {0,1,0,0,1,-4,1,0,0,1,0,0,0,0,0,0},
    {0,0,1,0,0,1,-4,1,0,0,1,0,0,0,0,0},
    {0,0,0,1,0,0,1,-4,0,0,0,1,0,0,0,0},
    {0,0,0,0,1,0,0,0,-4,1,0,0,1,0,0,0},
    {0,0,0,0,0,1,0,0,1,-4,1,0,0,1,0,0},
    {0,0,0,0,0,0,1,0,0,1,-4,1,0,0,1,0},
    {0,0,0,0,0,0,0,1,0,0,1,-4,0,0,0,1},
    {0,0,0,0,0,0,0,0,1,0,0,0,-4,1,0,0},
    {0,0,0,0,0,0,0,0,0,1,0,0,1,-4,1,0},
    {0,0,0,0,0,0,0,0,0,0,1,0,0,1,-4,1},
    {0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,-4},
  }

  local b = {-11, -3, -3, -11, -8, 0, 0, -8, -8, 0, 0, -8, -10,-2, -2, -10}

  local x = sor(m, b)
  assert(fuzzy_eq(x[1], 5.454459))
  assert(fuzzy_eq(x[2], 4.594688))
  assert(fuzzy_eq(x[3], 4.594679))
  assert(fuzzy_eq(x[4], 5.454531))
  assert(fuzzy_eq(x[5], 6.223469))
  assert(fuzzy_eq(x[6], 5.329580))
  assert(fuzzy_eq(x[7], 5.329561))
  assert(fuzzy_eq(x[8], 6.223491))
  assert(fuzzy_eq(x[9], 6.109833))
  assert(fuzzy_eq(x[10], 5.170488))
  assert(fuzzy_eq(x[11], 5.170455))
  assert(fuzzy_eq(x[12], 6.109845))
  assert(fuzzy_eq(x[13], 5.045460))
  assert(fuzzy_eq(x[14], 4.071980))
  assert(fuzzy_eq(x[15], 4.071964))
  assert(fuzzy_eq(x[16], 5.045457))
end)

print(('-'):rep(80))
print(('Total : %02d: Pass: %02d - Failed : %02d - Success: %.2f %%')
  :format(total, pass, total-pass, (pass*100/total)))