woggioni/mmath
1
0
Fork 0
Code Packages Releases Activity

added rational numbers

Browse Source
This commit is contained in:
Walter Oggioni
2023-05-24 22:47:16 +08:00
parent d4f64d25a0
commit 1e580cd6da
6 changed files with 278 additions and 51 deletions
Download Patch File Download Diff File Expand all files Collapse all files

28
src/mmath/hmatrix.nim
View File

@@ -42,7 +42,7 @@ proc newHMatrix*[T](rows, columns : int, values : openarray[T]) : HMatrix[T] =
proc identity*[T](sz : int) : HMatrix[T] =
result = newHMatrix[T](sz,sz,0)
for i in 0...sz:
result[i,i] = 1
result[i,i] = T.one
proc `$`*[T](m : HMatrix[T]) : string =
result = "["
@@ -60,7 +60,7 @@ proc `$`*[T](m : HMatrix[T]) : string =
else: result &= ", "
proc `*`*[T](m1 : HMatrix[T], m2 : HMatrix[T]) : HMatrix[T] =
result = newHMatrix[T](m1.rows, m2.columns, 0)
result = newHMatrix[T](m1.rows, m2.columns, T.zero)
for i in 0...result.rows:
for j in 0...result.columns:
for k in 0...m1.columns:
@@ -239,7 +239,7 @@ proc det*[T](m : HMatrix[T]) : T =
raise newException(SizeError, "Matrix must be square in order to compute the determinant")
var clone = m.clone()
clone.gauss_jordan_low()
result = 1
result = T.one
for i in 0...clone.rows:
result *= clone[i, i]
@@ -256,7 +256,7 @@ proc invert*[T](m : HMatrix[T]) : HMatrix[T] =
result[i, j] /= f
proc triu*[T](m : HMatrix[T], diag_replace: T) : HMatrix[T] =
result = newHMatrix[T](m.rows, m.columns, 0)
result = newHMatrix[T](m.rows, m.columns, T.zero)
for i in 0...m.rows:
for j in i...m.columns:
if i == j:
@@ -265,14 +265,14 @@ proc triu*[T](m : HMatrix[T], diag_replace: T) : HMatrix[T] =
result[i, j] = m[i, j]
proc triu*[T](m : HMatrix[T]) : HMatrix[T] =
result = newHMatrix[T](m.rows, m.columns, 0)
result = newHMatrix[T](m.rows, m.columns, T.zero)
for i in 0...m.rows:
for j in i...m.columns:
result[i, j] = m[i, j]
proc tril*[T](m : HMatrix[T], diag_replacement : T) : HMatrix[T] =
result = newHMatrix[T](m.rows, m.columns, 0)
result = newHMatrix[T](m.rows, m.columns, T.zero)
for i in 0...m.rows:
for j in 0...(i + 1):
if i == j:
@@ -281,13 +281,13 @@ proc tril*[T](m : HMatrix[T], diag_replacement : T) : HMatrix[T] =
result[i, j] = m[i, j]
proc tril*[T](m : HMatrix[T]) : HMatrix[T] =
result = newHMatrix[T](m.rows, m.columns, 0)
result = newHMatrix[T](m.rows, m.columns, T.zero)
for i in 0...m.rows:
for j in 0...(i + 1):
result[i, j] = m[i, j]
proc lu_row[T](m : var HMatrix[T], i : int) =
if m[i, i] == 0:
if m[i, i] == T.zero:
raise newException(SingularMatrixError, "Matrix is singular")
for j in i...m.columns:
for k in 0...i:
@@ -345,9 +345,9 @@ proc lu_invert*[T](m : HMatrix[T], pivot : HPivot[T]) : HMatrix[T] =
for i in 0...m.rows:
for j in 0...m.rows:
if pivot[j] == i:
result[j, i] = 1
result[j, i] = T.one
else:
result[j, i] = 0
result[j, i] = T.zero
for k in range(j):
result[j, i] -= m[j, k] * result[k, i]
for j in m.rows-->0:
@@ -361,11 +361,11 @@ proc lu_det*[T](m : var HMatrix[T]) : T =
if m.rows != m.columns:
raise newException(SizeError, "Matrix must be square in order to compute the determinant")
let pivot = m.lup()
result = 1
result = T.one
for i in 0...m.rows:
result *= m[i, i]
if pivot.permutations mod 2 != 0:
result *= -1
result *= -T.one
proc lu_det*[T](m : HMatrix[T]) : T =
if m.rows != m.columns:
@@ -374,7 +374,7 @@ proc lu_det*[T](m : HMatrix[T]) : T =
lu_det(clone)
proc squared_norm2*[T](m : HMatrix[T]): T =
result = T(0)
result = T.zero
for i, j, v in items(m):
result += v * v
@@ -394,5 +394,5 @@ proc `*`*[T](pivot : HPivot[T], m : HMatrix[T]) : HMatrix[T] =
proc from_pivot*[T](pivot : HPivot[T]): HMatrix[T] =
result = newHMatrix[T](len(pivot), len(pivot))
for i in 0...pivot.len:
result[pivot[i], i] = 1
result[pivot[i], i] = T.one

118
src/mmath/rational.nim Normal file
View File

@@ -0,0 +1,118 @@
proc one*[T : SomeNumber](_: typedesc[T]) : T = 1.T
proc zero*[T : SomeNumber](_: typedesc[T]) : T = 0.T
proc gcd*[T : SomeInteger](a : T, b : T) : T =
var n1 = a
var n2 = b
var tmp : T
while n2 != T.zero:
tmp = n1
n1 = n2
n2 = tmp mod n2
return n1
proc mcm*[T : SomeInteger](n1 : T, n2 : T) : T = n1 * n2 div gcd(n1, n2)
proc pow*[T : SomeInteger](n : T, e : T) : T =
result = T.one
for i in 0..<e:
result *= n
type Rational*[T : SomeInteger] = object
num* : T
den* : T
proc simplify*[T](self : var Rational[T]) : void =
let gcd = gcd(self.num.abs(), self.den.abs())
let invertSign = self.den < T.zero
let num = self.num div gcd * (if invertSign: (T.zero - T.one) else : T.one)
let den = self.den div gcd * (if invertSign: (T.zero - T.one) else : T.one)
self.num = num
self.den = den
proc newRational*[T](num : T) : Rational[T] = Rational[T](num: num, den: T(1))
proc newRational*[T](num : T, den: T) : Rational[T] = Rational[T](num: num, den: den)
proc zero*[T](_: typedesc[Rational[T]]): Rational[T] = newRational(T.zero, T.one)
proc one*[T](_: typedesc[Rational[T]]): Rational[T] = newRational(T.one, T.one)
proc abs*[T](self : Rational[T]): Rational[T] = Rational[T](num: self.num.abs(), den: self.den.abs())
proc pow*[T](self : Rational[T], e : SomeInteger): Rational[T] = newRational(self.num.pow(e), self.den.pow(e))
proc sqrt*[T](self : Rational[T]) : Rational[T] = newRational(sqrt(self.num), sqrt(self.den))
proc `-`*[T](r1 : Rational[T]) : Rational[T] = newRational(-r1.num, r1.den)
proc `+`*[T](r1 : Rational[T], r2: Rational[T]) : Rational[T] =
let den = mcm(r1.den, r2.den)
result = newRational(r1.num * den div r1.den + r2.num * den div r2.den, den)
result.simplify()
proc `-`*[T](r1 : Rational[T], r2: Rational[T]) : Rational[T] =
let den = mcm(r1.den, r2.den)
result = newRational(r1.num * den div r1.den - r2.num * den div r2.den, den)
result.simplify()
proc `*`*[T](r1 : Rational[T], r2: Rational[T]) : Rational[T] =
result = newRational(r1.num * r2.num, r1.den * r2.den)
result.simplify()
proc `/`*[T](r1 : Rational[T], r2: Rational[T]) : Rational[T] =
result = newRational(r1.num * r2.den, r1.den * r2.num)
result.simplify()
proc `+=`*[T](r1 : var Rational[T], r2: Rational[T]) : void =
let den = mcm(r1.den, r2.den)
r1.num = r1.num * den div r1.den + r2.num * den div r2.den
r1.den = den
r1.simplify()
proc `-=`*[T](r1 : var Rational[T], r2: Rational[T]) : void =
let den = mcm(r1.den, r2.den)
r1.num = r1.num * den div r1.den - r2.num * den div r2.den
r1.den = den
r1.simplify()
proc `*=`*[T](r1 : var Rational[T], r2: Rational[T]) : void =
r1.num = r1.num * r2.num
r1.den = r1.den * r2.den
r1.simplify()
proc `/=`*[T](r1 : var Rational[T], r2: Rational[T]) : void =
r1.num = r1.num * r2.den
r1.den = r1.den * r2.num
r1.simplify()
proc `cmp`*[T](r1 : Rational[T], r2 : Rational[T]) : int =
cmp(r1.num * r2.den, r1.den * r2.num)
proc `==`*[T](r1 : Rational[T], r2 : Rational[T]) : bool =
cmp(r1, r2) == 0
proc `<`*[T](r1 : Rational[T], r2 : Rational[T]) : bool =
cmp(r1, r2) < 0
proc `<=`*[T](r1 : Rational[T], r2 : Rational[T]) : bool =
cmp(r1, r2) <= 0
proc `>`*[T](r1 : Rational[T], r2 : Rational[T]) : bool =
cmp(r1, r2) > 0
proc `>=`*[T](r1 : Rational[T], r2 : Rational[T]) : bool =
cmp(r1, r2) >= 0
proc `$`*[T](r : Rational[T]): string =
let zero = T.zero
let one = T.one
if r.num == T.zero and r.den != zero:
result = $zero
else:
let negative = r.num == r.num.abs() xor r.den == r.den.abs()
if r.den.abs() == one:
result = $(if negative: (zero - one) * r.num else: r.num)
else:
result = (if negative: "-" else: "") & $(r.num.abs()) & "/" & $(r.den.abs())

57
src/mmath/smatrix.nim
View File

@@ -38,7 +38,7 @@ proc newSMatrixFromArray*[ROWS, COLUMNS : static[int], T](values : array[0..(ROW
proc identity*[SIZE: static[int], T]() : SquareSMatrix[SIZE, T] =
for i in 0...SIZE:
result[i,i] = 1
result[i,i] = T.one
proc `$`*[ROWS, COLUMNS : static[int], T](m : SMatrix[ROWS, COLUMNS, T]) : string =
result = "["
@@ -60,8 +60,9 @@ proc `*`*[ROWS1, COLUMNS2, COMMON : static[int], T](
m2 : SMatrix[COMMON, COLUMNS2, T]) : SMatrix[ROWS1, COLUMNS2, T] =
for i in 0...result.rows:
for j in 0...result.columns:
result[i, j] = T.zero
for k in 0...m1.columns:
result[i, j] = result[i, j] + m1[i, k] * m2[k, j]
result[i, j] += m1[i, k] * m2[k, j]
proc `*`*[SIZE : static[int], T](v : SVector[SIZE, T], m : SquareSMatrix[SIZE, T]) : SVector[SIZE, T] =
for i in 0...m.rows:
@@ -125,7 +126,7 @@ proc `==`*[ROWS, COLUMNS : static[int], T](m1 : SMatrix[ROWS, COLUMNS, T], m2 :
return true
proc squared_norm2*[ROWS, COLUMNS : static[int], T](m : SMatrix[ROWS, COLUMNS, T]): T =
result = T(0)
result = T.zero
for i, j, v in items(m):
result += v * v
@@ -182,7 +183,7 @@ proc gauss_jordan_low*[ROWS, COLUMNS : static[int], T](
other : var SMatrix[ROWS, COLUMNS, T]) =
var pivot = newSPivot[ROWS, T]()
for i in 0...m.rows:
if m[i, i] == 0:
if m[i, i] == T.zero:
for j in (i + 1)...m.columns:
if m[j, i] != 0:
m.swap_rows(i, j, pivot, other)
@@ -238,7 +239,7 @@ proc gauss_jordan_high*[ROWS, COLUMNS : static[int], T](
proc det*[SIZE : static[int], T](m : SquareSMatrix[SIZE, T]) : T =
var clone = m.clone()
clone.gauss_jordan_low()
result = 1
result = T.one
for i in 0...clone.rows:
result *= clone[i, i]
@@ -252,34 +253,44 @@ proc invert*[SIZE : static[int], T](m : SquareSMatrix[SIZE, T]) : SquareSMatrix[
for j in 0...result.columns:
result[i, j] /= f
proc triu*[SIZE : static[int], T](m : SquareSMatrix[SIZE, T], diag_replace: T) : SquareSMatrix[SIZE, T] =
proc triu*[SIZE : static[int], T](m : SquareSMatrix[SIZE, T], diag_replacement: T) : SquareSMatrix[SIZE, T] =
for i in 0...m.rows:
for j in i...m.columns:
if i == j:
result[i, j] = diag_replace
else:
for j in 0...m.columns:
if i < j:
result[i, j] = m[i, j]
elif i == j:
result[i, j] = diag_replacement
else:
result[i, j] = T.zero
proc triu*[SIZE : static[int], T](m : SquareSMatrix[SIZE, T]) : SquareSMatrix[SIZE, T] =
for i in 0...m.rows:
for j in i...m.columns:
result[i, j] = m[i, j]
for j in 0...m.columns:
if i <= j:
result[i, j] = m[i, j]
else:
result[i, j] = T.zero
proc tril*[SIZE : static[int], T](m : SquareSMatrix[SIZE, T], diag_replacement : T) : SquareSMatrix[SIZE, T] =
for i in 0...m.rows:
for j in 0...(i + 1):
if i == j:
for j in 0...m.columns:
if i > j:
result[i, j] = m[i, j]
elif i == j:
result[i, j] = diag_replacement
else:
result[i, j] = m[i, j]
result[i, j] = T.zero
proc tril*[SIZE : static[int], T](m : SquareSMatrix[SIZE, T]) : SquareSMatrix[SIZE, T] =
for i in 0...m.rows:
for j in 0...(i + 1):
result[i, j] = m[i, j]
for j in 0...m.columns:
if i >= j:
result[i, j] = m[i, j]
else:
result[i, j] = T.zero
proc lu_row[SIZE : static[int], T](m : var SquareSMatrix[SIZE, T], i : int) =
if m[i, i] == 0:
if m[i, i] == T.zero:
raise newException(SingularMatrixError, "Matrix is singular")
for j in i...m.columns:
for k in 0...i:
@@ -334,9 +345,9 @@ proc lu_invert*[SIZE : static[int], T](m : SquareSMatrix[SIZE, T], pivot : SPivo
for i in 0...m.rows:
for j in 0...m.rows:
if pivot[j] == i:
result[j, i] = 1
result[j, i] = T.one
else:
result[j, i] = 0
result[j, i] = T.zero
for k in range(j):
result[j, i] -= m[j, k] * result[k, i]
for j in m.rows-->0:
@@ -348,11 +359,11 @@ proc lu_invert*[SIZE : static[int], T](m : SquareSMatrix[SIZE, T]) : SquareSMatr
proc lu_det*[SIZE : static[int], T](m : var SquareSMatrix[SIZE, T]) : T =
let pivot = m.lup()
result = 1
result = T.one
for i in 0...m.rows:
result *= m[i, i]
if pivot.permutations mod 2 != 0:
result *= -1
result *= -T.one
proc lu_det*[SIZE : static[int], T](m : SquareSMatrix[SIZE, T]) : T =
var clone = m.clone()
@@ -371,5 +382,5 @@ proc `*`*[ROWS, COLUMNS : static[int], T](pivot : SPivot[ROWS, T], m : SMatrix[R
proc from_pivot*[SIZE : static[int], T](pivot : SPivot[SIZE, T]): SquareSMatrix[SIZE, T] =
result = newSMatrix[T](len(pivot), len(pivot))
for i in 0...SIZE:
result[pivot[i], i] = 1
result[pivot[i], i] = T.one

13
tests/test_hmatrix.nim
View File

@@ -1,7 +1,8 @@
from random import initRand, rand
from nwo/utils import `...`
from mmath/rational import newRational, Rational, one, zero, abs, `>`, `<`, `==`, `*`, `+`, `-`, `/`, `/=`, `+=`, `sqrt`
from mmath/hmatrix import det, lu, from_pivot, newHMatrix, HMatrix, lu_det, invert, identity,
clone, tril, triu, lu_solve, `*`, `-`, `+`, `+=`, `-=`, `==`, norm2, transpose, lup
clone, tril, triu, lu_solve, `*`, `-`, `+`, `+=`, `-=`, `==`, norm2, transpose, lup, squared_norm2
from mmath/hvector import buildHVector, newHVector, `-`, abs, norm
import unittest
@@ -58,16 +59,16 @@ suite "Nim linear algebra library":
test "LU decomposition":
var rng = initRand(101325)
var arr : array[0..(25 - 1), float32]
var arr : array[0..(25 - 1), Rational[int64]]
for i in 0..<arr.len:
arr[i] = rng.rand(-4f32..4f32)
let mtx = newHMatrix[float32](5, 5, arr)
arr[i] = newRational(rng.rand(-20..20).int64, 20.int64)
let mtx = newHMatrix[Rational[int64]](5, 5, arr)
var lu = mtx.clone()
let pivot = lu.lup()
let l = lu.tril(1.0)
let l = lu.tril(Rational[int64].one)
let u = lu.triu()
let err = pivot * mtx - (l * u)
check(err.norm2() < 1e-5)
check(err.squared_norm2() == Rational[int64].zero)
test "Linear system solve":
var rng = initRand(101325)

96
tests/test_rational.nim Normal file
View File

@@ -0,0 +1,96 @@
import unittest
from mmath/rational import newRational, `+`, `-`, `*`, `/`, `$`, simplify, gcd, pow, mcm, `==`, `cmp`, `<`, `>`, `<=`, `>=`, abs
suite "Nim rational numbers":
test "gcd":
let a = 14
let b = 21
check 7 == gcd(a, b)
test "mcm":
let a = 14
let b = 21
check 42 == mcm(a, b)
test "pow":
check 625 == pow(5, 4)
test "==":
let r1 = newRational(1, 3)
let r2 = newRational(5, 15)
let r3 = newRational(35, 105)
check r1 == r2
check r1 == r3
check r2 == r3
test "+":
let r1 = newRational(1, 3)
let r2 = newRational(2, 3)
let r3 = r1 + r2
check r3 == newRational(1)
test "-":
let r1 = newRational(1, 3)
let r2 = newRational(2, 3)
let r3 = r1 - r2
check r3 == newRational(-1, 3)
test "*":
let r1 = newRational(1, 3)
let r2 = newRational(2, 3)
let r3 = r1 * r2
check r3 == newRational(2, 9)
test "/":
let r1 = newRational(1, 3)
let r2 = newRational(2, 3)
let r3 = r1 / r2
check r3 == newRational(1, 2)
test "simplify":
var r1 = newRational(65, 169)
check r1 == newRational(5, 13)
r1.simplify()
check r1.num == 5
check r1.den == 13
test "pow":
let r1 = newRational(65, 169)
let r2 = r1.pow(2)
check r2 == newRational(25, 169)
test "cmp":
let r1 = newRational(65, 169)
let r2 = newRational(5, 13)
let r3 = newRational(6, 13)
check r1 < r3
check r3 > r1
check r1 <= r3
check r1 <= r2
check r3 >= r1
check r2 >= r1
check r2 == r1
test "abs":
let r1 = newRational(-65, -169)
let r2 = newRational(65, -169)
check r1 == r1.abs()
check r1 == r2.abs()
check r1 != r2
test "i64":
let p1 = 32452867i64
let p2 = 49979687i64
let p3 = 15485867i64
let r1 = newRational(p3, p1 * p2)
let r2 = newRational(p2, p1 * p3)
var r3 = newRational(p2, p2 * p3)
check newRational(1i64, p3) == r3
r3.simplify()
check r3.num == 1i64
check r3.den == p3
expect(OverflowDefect):
discard r1 / r2
expect(OverflowDefect):
discard r1 + r2

17
tests/test_smatrix.nim
View File

@@ -2,9 +2,10 @@ from random import initRand, rand
from nwo/utils import `...`
from mmath/smatrix import det, lu, lup, lu_det, lup, det, from_pivot, invert, identity,
newSMatrix, SMatrix, clone, tril, triu, lu_solve, `$`, `*`, `-`, `+`, `-=`, `+=`, `==`, `*`, norm2, newSMatrixFromArray,
gauss_jordan_high, gauss_jordan_low, transpose
gauss_jordan_high, gauss_jordan_low, transpose, squared_norm2
from mmath/svector import buildSVector, `-`, abs, norm, Svector
from mmath/pivot import SingularMatrixError
from mmath/rational import newRational, Rational, one, zero, abs, `>`, `<`, `==`, `*`, `+`, `-`, `/`, `/=`, `+=`, `sqrt`
import unittest
suite "Nim linear algebra library":
@@ -58,19 +59,19 @@ suite "Nim linear algebra library":
test "Inverse":
let err = test_matrix * test_matrix.invert() - identity[3, float32]()
check(err.norm2() < 1e-5)
test "LU decomposition":
var rng = initRand(101325)
var arr : array[0..(25 - 1), float32]
var arr : array[0..(25 - 1), Rational[int64]]
for i in 0..<arr.len:
arr[i] = rng.rand(-4f32..4f32)
let mtx = newSMatrixFromArray[5, 5, float32](arr)
var lu = mtx.clone()
arr[i] = newRational(rng.rand(-20..20).int64, 20.int64)
let mtx = newSMatrixFromArray[5, 5, Rational[int64]](arr)
var lu = mtx.clone()
let pivot = lu.lup()
let l = lu.tril(1.0)
let l = lu.tril(Rational[int64].one)
let u = lu.triu()
let err = pivot * mtx - (l * u)
check(err.norm2() < 1e-5)
check(err.squared_norm2() == Rational[int64].zero)
test "Linear system solve":
var rng = initRand(101325)