Python Code - Gauss's Circle Problem

This code uses the algorithm discussed here.

Iterative version of the code - Python

from math import sqrt, ceil, floor
from timeit import default_timer

start = default_timer()

def matmult (A, B):
    rows_A = len(A)
    cols_A = len(A[0])
    rows_B = len(B)
    cols_B = len(B[0])

    if cols_A != rows_B:
      print("Cannot multiply the two matrices. Incorrect dimensions.")
      return

    # Create the result matrix
    # Dimensions would be rows_A x cols_B
    C = [[0 for row in range(cols_B)] for col in range(rows_A)]

    for i in range(rows_A):
        for j in range(cols_B):
            for k in range(cols_A):
                C[i][j] += A[i][k] * B[k][j]
    return C

def gauss(n):
    sn = ceil(sqrt(n))
    res = 0
    stk = [([[1, 0, 0], [0, 1, 0], [0, 0, -n]], sn, sn)]
    while stk:
        m, w, h = stk.pop()
        if w <= 0 or h <= 0:
            continue
        a, b, c, d, e, f = m[0][0], 2 * m[0][1], m[1][1], 2 * m[0][2], 2 * m[1][2], m[2][2]
        if d * d - 4 * a * f < 0 or e * e - 4 * c * f < 0:
            res += w * h
            continue
        x0, y0 = (- d + sqrt(d * d - 4 * a * f)) / 2 / a, (- e + sqrt(e * e - 4 * c * f)) / 2 / c
        xint, yint, uint, vint, flag = 0, 0, 0, 0, 0
        if floor(x0) == ceil(x0):
            xint = 1
        if floor(y0) == ceil(y0):
            yint = 1
        if w <= 1:
            res += (h - ceil(y0)) * w - yint
            continue
        if h <= 1:
            res += (w - ceil(x0)) * h - xint
            continue
        if w - x0 > 1:
            res += (w - ceil(x0)) * h - xint
            stk.append((m, ceil(x0), h))
            continue
        if h - y0 > 1:
            res += (h - ceil(y0)) * w - yint
            stk.append((m, w, ceil(y0)))
            continue
        temp = (a - b + c) * (c * d * d - b * d * e + a * e * e + b * b * f - 4 * a * c * f)
        if temp < 0:
            continue
        u = ((a - b + c) * (b * e - 2 * c * d) - (b - 2 * c) * sqrt(temp)) / (a - b + c) / (4 * a * c - b * b)
        if floor(u) == ceil(u):
            uint = 1
        if u > w:
            u, flag = w, 1
        if u < 0:
            u, flag = 0, 1
        temp = (e  + b * u) * (e + b * u) - 4 * c * (f + u * (d + a * u))
        if temp < 0:
            continue
        v = (- e - b * u + sqrt(temp)) / 2 / c
        if floor(v) == ceil(v):
            vint = 1
        if v < 0:
            v = 0
        u, v = ceil(u), ceil(v)
        if u == w and v == h:
            continue
        if uint == 1 and vint == 1 and flag == 0:
            res -= 1
        res += (w - u + h - 1 - v) * (w - u + h - v) // 2
        stk.append((matmult(matmult([[1, 0, 0], [-1, 1, 0], [h - v, v, 1]], m), [[1, -1, h - v], [0, 1, v], [0, 0, 1]]), u - (h - v), h - v))
        stk.append((matmult(matmult([[1, -1, 0], [0, 1, 0], [u, w - u, 1]], m), [[1, 0, u], [-1, 1, w - u], [0, 0, 1]]), w - u, v - (w - u)))
    res = (sn - 1) * (sn - 1) - res
    res += sn - 1
    res *= 4
    res += 1
    if sn * sn == n:
        res += 4
    return res

print(gauss(10 ** 20))
print(default_timer() - start)


Recursive version of the code - Python

from math import sqrt, ceil, floor
from timeit import default_timer

start = default_timer()

def matmult (A, B):
    rows_A = len(A)
    cols_A = len(A[0])
    rows_B = len(B)
    cols_B = len(B[0])

    if cols_A != rows_B:
      print("Cannot multiply the two matrices. Incorrect dimensions.")
      return

    # Create the result matrix
    # Dimensions would be rows_A x cols_B
    C = [[0 for row in range(cols_B)] for col in range(rows_A)]

    for i in range(rows_A):
        for j in range(cols_B):
            for k in range(cols_A):
                C[i][j] += A[i][k] * B[k][j]
    return C

def poins(m, w, h):
    res = 0
    if w <= 0 or h <= 0:
        return 0
    a, b, c, d, e, f = m[0][0], 2 * m[0][1], m[1][1], 2 * m[0][2], 2 * m[1][2], m[2][2]
    if d * d - 4 * a * f < 0 or e * e - 4 * c * f < 0:
        return w * h
    x0, y0 = (- d + sqrt(d * d - 4 * a * f)) / 2 / a, (- e + sqrt(e * e - 4 * c * f)) / 2 / c
    xint, yint, uint, vint, flag = 0, 0, 0, 0, 0
    if floor(x0) == ceil(x0):
        xint = 1
    if floor(y0) == ceil(y0):
        yint = 1
    if w <= 1:
        return (h - ceil(y0)) * w - yint
    if h <= 1:
        return (w - ceil(x0)) * h - xint
    if w - x0 > 1:
        return (w - ceil(x0)) * h - xint + poins(m, ceil(x0), h)
    if h - y0 > 1:
        return (h - ceil(y0)) * w - yint + poins(m, w, ceil(y0))
    temp = (a - b + c) * (c * d * d - b * d * e + a * e * e + b * b * f - 4 * a * c * f)
    if temp < 0:
        return 0
    u = ((a - b + c) * (b * e - 2 * c * d) - (b - 2 * c) * sqrt(temp)) / (a - b + c) / (4 * a * c - b * b)
    if floor(u) == ceil(u):
        uint = 1
    if u > w:
        u, flag = w, 1
    if u < 0:
        u, flag = 0, 1
    temp = (e  + b * u) * (e + b * u) - 4 * c * (f + u * (d + a * u))
    if temp < 0:
        return 0
    v = (- e - b * u + sqrt(temp)) / 2 / c
    if floor(v) == ceil(v):
        vint = 1
    if v < 0:
        v = 0
    if uint == 1 and vint == 1 and flag == 0:
        res -= 1
    u, v = ceil(u), ceil(v)
    if u == w and v == h:
        return 0
    res += (w - u + h - 1 - v) * (w - u + h - v) // 2
    res += poins(matmult(matmult([[1, 0, 0], [-1, 1, 0], [h - v, v, 1]], m), [[1, -1, h - v], [0, 1, v], [0, 0, 1]]), u - (h - v), h - v)
    res += poins(matmult(matmult([[1, -1, 0], [0, 1, 0], [u, w - u, 1]], m), [[1, 0, u], [-1, 1, w - u], [0, 0, 1]]), w - u, v - (w - u))
    #print(m, w, h, u, v, res)
    return res

def gauss(n):
    m = ceil(sqrt(n))
    res = poins([[1, 0, 0], [0, 1, 0], [0, 0, -n]], m, m)
    res = (m - 1) * (m - 1) - res
    res += m - 1
    res *= 4
    res += 1
    if m * m == n:
        res += 4
    return res

print(gauss(10 ** 14))

print(default_timer() - start)


Recursive version of the code


Clear["Global`*"];
Poins[m0_, w0_, h0_] :=
  Poins[m0, w0, h0] =
   Block[{$RecursionLimit = Infinity, m = m0, w = w0, h = h0, a, b, c,
      d, e, f, temp, u, v, res = 0, trans, x0, y0, Flag = 0},
    If[Or[w <= 0, h <= 0], Return[0];];
    If[w < h,
     temp = {{0, 1, 0}, {1, 0, 0}, {0, 0, 1}};
     m = Transpose[temp].m.temp;
     temp = w; w = h; h = temp;
     ];
    a = m[[1, 1]]; b = 2 m[[1, 2]]; c = m[[2, 2]]; d = 2 m[[1, 3]];
    e = 2 m[[2, 3]]; f = m[[3, 3]];
    If[Or[d*d - 4*a*f < 0, e*e - 4*c*f < 0], Return[w h];];
    x0 = (-d + Sqrt[d*d - 4*a*f])/2/a;
    y0 = (-e + Sqrt[e*e - 4*c*f])/2/c;
    If[w <= 1, Return[(h - Ceiling[y0]) w - Boole[IntegerQ[y0]]];];
    If[h <= 1, Return[(w - Ceiling[x0]) h - Boole[IntegerQ[x0]]];];
    If[w - x0 > 1,
     Return[(w - Ceiling[x0]) h - Boole[IntegerQ[x0]] +
        Poins[m, Ceiling[x0], h]];
     ];
    If[h - y0 > 1,
     Return[(h - Ceiling[y0]) w - Boole[IntegerQ[y0]] +
        Poins[m, w, Ceiling[y0]]];
     ];
    temp = (a - b + c)*(c*d*d - b*d*e + a*e*e + b*b*f - 4*a*c*f);
    If[temp < 0, Return[0];];
    u = ((a - b + c)*(b*e - 2*c*d) - (b - 2*c)*Sqrt[temp])/(a - b +
         c)/(4*a*c - b*b);
    If[u > w, u = w; Flag = 1;];
    If[u < 0, u = 0; Flag = 1;];
    temp = (e + b*u)*(e + b*u) - 4*c*(f + u*(d + a*u));
    If[temp < 0, Return[0];];
    v = (-e - b*u + Sqrt[temp])/2/c;
    If[v < 0, v = 0;, v = Ceiling[v];];
    If[And[IntegerQ[u], IntegerQ[v], Flag == 0], res -= 1;];
    u = Ceiling[u]; v = Ceiling[v];
    If[And[u == w, v == h], Return[0];];
    res += (w - u + h - 1 - v)*(w - u + h - v)/2;
    trans = {{1, -1, h - v}, {0, 1, v}, {0, 0, 1}};
    res += Poins[Transpose[trans].m.trans, u - (h - v), h - v];
    trans = {{1, 0, u}, {-1, 1, w - u}, {0, 0, 1}};
    res += Poins[Transpose[trans].m.trans, w - u, v - (w - u)];
    res
    ];
GaussCnt[n0_] := Block[{n = n0, m, res},
   m = Ceiling[Sqrt[n]];
   res = Poins[{{1, 0, 0}, {0, 1, 0}, {0, 0, -n}}, m, m];
   res = (m - 1)*(m - 1) - res;
   res += m - 1; res *= 4; res += 1;
   If[m m == n, res += 4;];
   res
   ];
Rad = Power[10, 6];
GaussCnt[Rad*Rad] // AbsoluteTiming



Note that in all the codes, the core function takes as input the square of the radius of the circle under consideration.



Until then
Yours Aye
Me