# 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):

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

a=(1,0,0,1)b=(0,1,0,1)c=(0,0,1,1)d=(0,1,1,1) \begin{aligned} &amp; a = ( 1, 0, 0, 1 ) \\ &amp; b = ( 0, 1, 0, 1 ) \\ &amp; c = ( 0, 0, 1, 1 ) \\ &amp; 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)​

118 篇文章22 人订阅

0 条评论