Gram-Schmidt 正交化的简单实现
Gram-Schmidt(格拉姆-施密特) 正交化可以正交化一组给定的向量,使这些向量两两垂直,这里列出一份简单的实现(Lua):
代码语言：javascript复制-- vector add
function add(a, b)
if a and b and #a == #b then
local ret = {}
for i = 1, #a do
table.insert(ret, a[i] b[i])
end
return ret
end
end
-- vector sub
function sub(a, b)
if a and b and #a == #b then
local ret = {}
for i = 1, #a do
table.insert(ret, a[i] - b[i])
end
return ret
end
end
-- dot product
function dot(a, b)
if a and b and #a == #b then
local ret = 0
for i = 1, #a do
ret = ret a[i] * b[i]
end
return ret
end
end
-- magnitude
function mag(a)
local val = dot(a, a)
if val then
return math.sqrt(val)
end
end
-- normalize, do not change param
function norm(a)
local magnitude = mag(a)
if magnitude and magnitude ~= 0 then
local normalize = {}
for i = 1, #a do
table.insert(normalize, a[i] / magnitude)
end
return normalize
end
end
-- project a to b
function proj(a, b)
if a and b and #a == #b then
local norm_b = norm(b)
local val = dot(a, norm_b)
if val then
local projection = {}
for i = 1, #norm_b do
table.insert(projection, norm_b[i] * val)
end
return projection
end
end
end
-- perpendicular a to b
function perp(a, b)
local projection = proj(a, b)
if projection then
return sub(a - projection)
end
end
-- gram schmidt
function gram_schmidt(...)
local vecs = { ... }
local ret = {}
if #vecs > 0 then
table.insert(ret, vecs[1])
end
for i = 2, #vecs do
local base = vecs[i]
for j = 1, i - 1 do
base = sub(base, proj(vecs[i], vecs[j]))
end
table.insert(ret, base)
vecs[i] = base
end
return table.unpack(ret)
end
-- use to check gram schmidt result
function check_perp(...)
local vecs = { ... }
for i = 1, #vecs - 1 do
for j = i 1, #vecs do
local val = dot(vecs[i], vecs[j])
if math.abs(val) > 0.001 then
return false
end
end
end
return true
end有兴趣的朋友可以试试这组向量的 Gram-Schmidt 正交化:
a=(1,0,0,1)b=(0,1,0,1)c=(0,0,1,1)d=(0,1,1,1) begin{aligned} & a = ( 1, 0, 0, 1 ) \ & b = ( 0, 1, 0, 1 ) \ & c = ( 0, 0, 1, 1 ) \ & d = ( 0, 1, 1, 1 ) end{aligned} a=(1,0,0,1)b=(0,1,0,1)c=(0,0,1,1)d=(0,1,1,1)
