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local function isreal(x)
return x == x
end
local function isnan(x)
return not x == x
end
local function roots(c)
local tol=1e-12
c[torch.lt(torch.abs(c),tol)]=0
local nonzero = torch.ne(c,0)
if nonzero:max() == 0 then
return 0
end
-- first non-zero
local _,pos = torch.max(nonzero,1)
pos = pos[1]
c=c[{ {pos,-1} }]
local nz = 0
for i=c:size(1),1,-1 do
if c[i] ~= 0 then
break
else
nz = nz + 1
end
end
c=c[{ {1,c:size(1)-nz} }]
local n = c:size(1)-1
if n == 1 then
local e = c.new({{-c[2]/c[1], 0}})
if nz > 0 then
return torch.cat(e, c.new(nz, 2):zero(), 1)
else
return e
end
elseif n > 1 then
local A = torch.diag(c.new(n-1):fill(1),-1)
A[1] = -c[{ {2,n+1} }]/c[1];
local e = torch.eig(A,'N')
if nz > 0 then
return torch.cat(e, c.new(nz,2):zero(), 1)
else
return e
end
else
return c.new(nz,2):zero()
end
end
local function real(x)
if type(x) == number then return x end
return x[{ {} , 1}]
end
local function imag(x)
if type(x) == 'number' then return 0 end
if x:nDimension() == 1 then
return x.new(x:size(1)):zero()
else
return x[{ {}, 2}]
end
end
local function polyval(p,x)
local pwr = p:size(1)
if type(x) == 'number' then
local val = 0
p:apply(function(pc) pwr = pwr-1; val = val + pc*x^pwr; return pc end)
return val
else
local val = x.new(x:size(1))
p:apply(function(pc) pwr = pwr-1; val:add(pc,torch.pow(x,pwr)); return pc end)
return val
end
end
----------------------------------------------------------------------
-- Minimum of interpolating polynomial based on function and
-- derivative values
--
-- ARGS:
-- points : N triplets (x,f,g), must be a Tensor
-- xmin : min value that brackets minimum (default: min of points)
-- xmax : max value that brackets maximum (default: max of points)
--
-- RETURN:
-- minPos : position of minimum
--
function optim.polyinterp(points,xminBound,xmaxBound)
-- locals
local sqrt = torch.sqrt
local mean = torch.mean
local max = math.max
local min = math.min
-- nb of points / order of polynomial
local nPoints = points:size(1)
local order = nPoints*2-1
-- returned values
local minPos
-- Code for most common case:
-- + cubic interpolation of 2 points w/ function and derivative values for both
-- + no xminBound/xmaxBound
if nPoints == 2 and order == 3 and not xminBound and not xmaxBound then
-- Solution in this case (where x2 is the farthest point):
-- d1 = g1 + g2 - 3*(f1-f2)/(x1-x2);
-- d2 = sqrt(d1^2 - g1*g2);
-- minPos = x2 - (x2 - x1)*((g2 + d2 - d1)/(g2 - g1 + 2*d2));
-- t_new = min(max(minPos,x1),x2);
local minVal,minPos = points[{ {},1 }]:min(1)
minVal = minVal[1] minPos = minPos[1]
local notMinPos = -minPos+3;
local d1 = points[{minPos,3}] + points[{notMinPos,3}]
- 3*(points[{minPos,2}]-points[{notMinPos,2}])
/ (points[{minPos,1}]-points[{notMinPos,1}]);
local d2 = sqrt(d1^2 - points[{minPos,3}]*points[{notMinPos,3}]);
if isreal(d2) then -- isreal()
local t = points[{notMinPos,1}] - (points[{notMinPos,1}]
- points[{minPos,1}]) * ((points[{notMinPos,3}] + d2 - d1)
/ (points[{notMinPos,3}] - points[{minPos,3}] + 2*d2))
minPos = min(max(t,points[{minPos,1}]),points[{notMinPos,1}])
else
minPos = mean(points[{{},1}])
end
return minPos
end
-- TODO: get the code below to work!
--error('<optim.polyinterp> extrapolation not implemented yet...')
-- Compute Bounds of Interpolation Area
local xmin = points[{{},1}]:min()
local xmax = points[{{},1}]:max()
xminBound = xminBound or xmin
xmaxBound = xmaxBound or xmax
-- Add constraints on function values
local A = points.new(nPoints*2,order+1):zero()
local b = points.new(nPoints*2,1):zero()
for i = 1,nPoints do
local constraint = points.new(order+1):zero()
for j = order,0,-1 do
constraint[order-j+1] = points[{i,1}]^j
end
A[i] = constraint
b[i] = points[{i,2}]
end
-- Add constraints based on derivatives
for i = 1,nPoints do
local constraint = points.new(order+1):zero()
for j = 1,order do
constraint[j] = (order-j+1)*points[{i,1}]^(order-j)
end
A[nPoints+i] = constraint
b[nPoints+i] = points[{i,3}]
end
-- Find interpolating polynomial
local res = torch.gels(b,A)
local params = res[{ {1,nPoints*2} }]:squeeze()
params[torch.le(torch.abs(params),1e-12)]=0
-- Compute Critical Points
local dParams = points.new(order):zero();
for i = 1,params:size(1)-1 do
dParams[i] = params[i]*(order-i+1)
end
-- nan/inf?
local nans = false
if torch.ne(dParams,dParams):max() > 0 or torch.eq(dParams,math.huge):max() > 0 then
nans = true
end
local cp = torch.cat(points.new{xminBound,xmaxBound},points[{{},1}])
if not nans then
local cproots = roots(dParams)
local cpi = points.new(cp:size(1),2):zero()
cpi[{ {1,cp:size(1)} , 1 }] = cp
cp = torch.cat(cpi,cproots,1)
end
-- Test Critical Points
local fmin = math.huge
-- Default to Bisection if no critical points valid:
minPos = (xminBound+xmaxBound)/2
for i = 1,cp:size(1) do
local xCP = cp[{ {i,i} , {} }]
local ixCP = imag(xCP)[1]
local rxCP = real(xCP)[1]
if ixCP == 0 and rxCP >= xminBound and rxCP <= xmaxBound then
local fCP = polyval(params,rxCP)
if fCP < fmin then
minPos = rxCP
fmin = fCP
end
end
end
return minPos,fmin
end
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