Generalized Fixed-Point Iteration using Gauss-Seidel and Jacobi Methods.ipynb

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   "cell_type": "markdown",
   "id": "6173c91c-5ebc-4ecd-a337-1b6a7899c141",
   "metadata": {},
   "source": [
    "# Function to perform Jacobi Iteration for fixed point iteration "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "bc2c777d-f7b2-4a52-b12e-fb8833a807a5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "read_input (generic function with 1 method)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using LinearAlgebra\n",
    "\n",
    "# Function to perform Jacobi Iteration and print all intermediate results\n",
    "function jacobi(A, b, x0; tol=1e-9, max_iter=1000)\n",
    "    n = length(b)               # Number of unknowns\n",
    "    x = copy(x0)                # Initial guess\n",
    "    x_new = zeros(n)            # Placeholder for updated values\n",
    "    iter = 0                    # Iteration counter\n",
    "    \n",
    "    while iter < max_iter\n",
    "        for i in 1:n\n",
    "            sum = 0.0\n",
    "            for j in 1:n\n",
    "                if i != j\n",
    "                    sum += A[i, j] * x[j]  # Sum of off-diagonal elements\n",
    "                end\n",
    "            end\n",
    "            x_new[i] = (b[i] - sum) / A[i, i]  # Update x[i] using the Jacobi formula\n",
    "        end\n",
    "        \n",
    "        # Print current iteration and solution\n",
    "        println(\"Iteration $iter: \", x_new)\n",
    "        \n",
    "        # Check for convergence (if the difference is below the tolerance)\n",
    "        if norm(x_new - x, Inf) < tol\n",
    "            println(\"Converged in $iter iterations\")\n",
    "            return x_new, iter\n",
    "        end\n",
    "        \n",
    "        # Update x for the next iteration\n",
    "        x = copy(x_new)\n",
    "        iter += 1\n",
    "    end\n",
    "    \n",
    "    println(\"Maximum iterations reached. Did not converge.\")\n",
    "    return x_new, iter\n",
    "end\n",
    "\n",
    "# Input function to take general matrix A, vector b, and initial guess x0\n",
    "function read_input()\n",
    "    # Reading matrix A dimensions\n",
    "    println(\"Enter the size of the matrix A (number of rows or columns, for a square matrix):\")\n",
    "    \n",
    "    # Input validation for matrix size\n",
    "    n = 0\n",
    "    while n <= 0\n",
    "        input = readline()\n",
    "        if isempty(input)\n",
    "            println(\"Input cannot be empty. Please enter a valid number:\")\n",
    "        else\n",
    "            try\n",
    "                n = parse(Int, input)\n",
    "                if n <= 0\n",
    "                    println(\"Size of matrix must be a positive integer. Please enter a valid number:\")\n",
    "                end\n",
    "            catch\n",
    "                println(\"Invalid input. Please enter a valid integer:\")\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "\n",
    "    # Initialize an empty matrix A and vector b\n",
    "    A = zeros(n, n)\n",
    "    b = zeros(n)\n",
    "\n",
    "    # Read matrix A with input validation\n",
    "    println(\"Enter the elements of matrix A row by row (space-separated values for each row):\")\n",
    "    for i in 1:n\n",
    "        row_valid = false\n",
    "        while !row_valid\n",
    "            input = readline()\n",
    "            row_values = split(input)\n",
    "            if length(row_values) == n\n",
    "                try\n",
    "                    A[i, :] = parse.(Float64, row_values)\n",
    "                    row_valid = true\n",
    "                catch\n",
    "                    println(\"Invalid input. Please enter valid floating point numbers for row $i:\")\n",
    "                end\n",
    "            else\n",
    "                println(\"Incorrect number of elements. Please enter $n elements for row $i:\")\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "\n",
    "    # Read vector b with input validation\n",
    "    println(\"Enter the elements of vector b (space-separated values):\")\n",
    "    vec_valid = false\n",
    "    while !vec_valid\n",
    "        input = readline()\n",
    "        vec_values = split(input)\n",
    "        if length(vec_values) == n\n",
    "            try\n",
    "                b = parse.(Float64, vec_values)\n",
    "                vec_valid = true\n",
    "            catch\n",
    "                println(\"Invalid input. Please enter valid floating point numbers for vector b:\")\n",
    "            end\n",
    "        else\n",
    "            println(\"Incorrect number of elements. Please enter $n elements for vector b:\")\n",
    "        end\n",
    "    end\n",
    "\n",
    "    # Read initial guess x0 with input validation\n",
    "    println(\"Enter the initial guess for vector x0 (space-separated values):\")\n",
    "    x0_valid = false\n",
    "    while !x0_valid\n",
    "        input = readline()\n",
    "        x0_values = split(input)\n",
    "        if length(x0_values) == n\n",
    "            try\n",
    "                x0 = parse.(Float64, x0_values)\n",
    "                x0_valid = true\n",
    "            catch\n",
    "                println(\"Invalid input. Please enter valid floating point numbers for vector x0:\")\n",
    "            end\n",
    "        else\n",
    "            println(\"Incorrect number of elements. Please enter $n elements for vector x0:\")\n",
    "        end\n",
    "    end\n",
    "\n",
    "    return A, b, x0\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "69e70d7c-0266-4b71-b3b9-59a7f7eb6f72",
   "metadata": {},
   "source": [
    "# Hard-coded Example Input (2x2 Matrix)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "9e27ff1c-5a93-4e8b-bdd3-2846daacfeb8",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0: [5.5, 1.8571428571428572]\n",
      "Iteration 1: [4.571428571428571, -2.0714285714285716]\n",
      "Iteration 2: [6.535714285714286, -1.408163265306122]\n",
      "Iteration 3: [6.204081632653061, -2.8112244897959187]\n",
      "Iteration 4: [6.905612244897959, -2.574344023323615]\n",
      "Iteration 5: [6.7871720116618075, -3.0754373177842567]\n",
      "Iteration 6: [7.037718658892128, -2.9908371511870055]\n",
      "Iteration 7: [6.9954185755935026, -3.1697990420658066]\n",
      "Iteration 8: [7.084899521032903, -3.139584696852502]\n",
      "Iteration 9: [7.069792348426251, -3.2034996578806454]\n",
      "Iteration 10: [7.1017498289403225, -3.1927088203044645]\n",
      "Iteration 11: [7.096354410152232, -3.21553559210023]\n",
      "Iteration 12: [7.1077677960501155, -3.211681721537308]\n",
      "Iteration 13: [7.105840860768654, -3.2198341400357964]\n",
      "Iteration 14: [7.109917070017898, -3.2184577576918962]\n",
      "Iteration 15: [7.109228878845948, -3.2213693357270707]\n",
      "Iteration 16: [7.110684667863535, -3.220877770604249]\n",
      "Iteration 17: [7.110438885302124, -3.221917619902525]\n",
      "Iteration 18: [7.110958809951263, -3.2217420609300893]\n",
      "Iteration 19: [7.110871030465045, -3.2221134356794727]\n",
      "Iteration 20: [7.111056717839737, -3.2220507360464614]\n",
      "Iteration 21: [7.111025368023231, -3.2221833698855256]\n",
      "Iteration 22: [7.111091684942763, -3.2221609771594513]\n",
      "Iteration 23: [7.111080488579725, -3.2222083463876876]\n",
      "Iteration 24: [7.111104173193844, -3.2222003489855178]\n",
      "Iteration 25: [7.111100174492758, -3.2222172665670317]\n",
      "Iteration 26: [7.111108633283516, -3.2222144103519708]\n",
      "Iteration 27: [7.111107205175985, -3.2222204523453684]\n",
      "Iteration 28: [7.111110226172684, -3.2222194322685604]\n",
      "Iteration 29: [7.11110971613428, -3.222221590123346]\n",
      "Iteration 30: [7.111110795061673, -3.2222212258101997]\n",
      "Iteration 31: [7.1111106129051, -3.222221996472624]\n",
      "Iteration 32: [7.111110998236312, -3.2222218663607856]\n",
      "Iteration 33: [7.111110933180393, -3.2222221415973658]\n",
      "Iteration 34: [7.111111070798683, -3.2222220951288523]\n",
      "Iteration 35: [7.111111047564426, -3.2222221934276303]\n",
      "Iteration 36: [7.111111096713815, -3.222222176831733]\n",
      "Iteration 37: [7.111111088415867, -3.222222211938439]\n",
      "Iteration 38: [7.111111105969219, -3.2222222060113332]\n",
      "Iteration 39: [7.111111103005666, -3.2222222185494425]\n",
      "Iteration 40: [7.111111109274721, -3.222222216432619]\n",
      "Iteration 41: [7.11111110821631, -3.2222222209105147]\n",
      "Iteration 42: [7.111111110455258, -3.2222222201545065]\n",
      "Iteration 43: [7.111111110077253, -3.2222222217537557]\n",
      "Iteration 44: [7.111111110876878, -3.222222221483752]\n",
      "Converged in 44 iterations\n",
      "Final Solution: [7.111111110876878, -3.222222221483752]\n",
      "Number of iterations: 44\n",
      "Time taken (seconds): 0.7620000839233398\n"
     ]
    }
   ],
   "source": [
    "# Hard-coded Example Input (2x2 Matrix)\n",
    "A_2x2 = [2.0  1.0;\n",
    "         5.0  7.0]\n",
    "\n",
    "b_2x2 = [11.0, 13.0]  # Vector b\n",
    "\n",
    "x0_2x2 = [0, 0]  # Initial guess, a vector of zeros (2 elements)\n",
    "\n",
    "\n",
    "\n",
    "function main()\n",
    "    # Start the timer\n",
    "    start_time = time()\n",
    "\n",
    "    # Call the Jacobi method with the matrix A_2x2, vector b_2x2, and initial guess x0_2x2\n",
    "    x_sol, iterations = jacobi(A_2x2, b_2x2, x0_2x2)\n",
    "\n",
    "    # End the timer\n",
    "    end_time = time()\n",
    "\n",
    "    # Calculate the duration\n",
    "    duration = end_time - start_time\n",
    "\n",
    "    # Output the final solution, number of iterations, and time taken\n",
    "    println(\"Final Solution: \", x_sol)\n",
    "    println(\"Number of iterations: \", iterations)\n",
    "    println(\"Time taken (seconds): \", duration)\n",
    "end\n",
    "\n",
    "# Call the main function\n",
    "main()\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "250ff29e-748f-49de-b72e-4215a901d83d",
   "metadata": {},
   "source": [
    "# Hard-coded Example Input (3x3 Matrix)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "af6ade92-5d1c-49bf-a63c-e948d5763ff0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0: [1.75, 1.3333333333333333, 1.2]\n",
      "Iteration 1: [1.7833333333333332, 1.1166666666666667, 0.31666666666666676]\n",
      "Iteration 2: [1.95, 1.7166666666666666, 0.39666666666666667]\n",
      "Iteration 3: [2.08, 1.7188888888888887, 0.12333333333333343]\n",
      "Iteration 4: [2.1488888888888886, 1.9444444444444444, 0.0964444444444446]\n",
      "Iteration 5: [2.2119999999999997, 1.9853333333333332, -0.007555555555555493]\n",
      "Iteration 6: [2.248222222222222, 2.0757037037037036, -0.036533333333333216]\n",
      "Iteration 7: [2.2780592592592592, 2.1070962962962962, -0.07992592592592587]\n",
      "Iteration 8: [2.2967555555555554, 2.1459703703703705, -0.09845037037037034]\n",
      "Iteration 9: [2.3111051851851854, 2.164552098765432, -0.1177392592592593]\n",
      "Iteration 10: [2.320572839506173, 2.1821945679012344, -0.12804187654320992]\n",
      "Iteration 11: [2.3275591111111114, 2.192218864197531, -0.13699239506172844]\n",
      "Iteration 12: [2.332302814814815, 2.2005146337448562, -0.1423993679012346]\n",
      "Iteration 13: [2.3357285004115225, 2.205700516872428, -0.1466664164609055]\n",
      "Iteration 14: [2.3380917333333335, 2.209687111111111, -0.14942590683127577]\n",
      "Iteration 15: [2.339778254485597, 2.212314515665295, -0.15149319111111126]\n",
      "Iteration 16: [2.3409519266941015, 2.2142548789026066, -0.1528814571632374]\n",
      "Iteration 17: [2.341784084016461, 2.2155716136735255, -0.15389233689986295]\n",
      "Iteration 18: [2.342365987643347, 2.2165229192720624, -0.1545854622727024]\n",
      "Iteration 19: [2.3427770953861913, 2.217178970729584, -0.15508236523749447]\n",
      "Iteration 20: [2.3430653339917695, 2.21764727528706, -0.15542700736907183]\n",
      "Iteration 21: [2.343268570664033, 2.217973116243304, -0.15567197691317797]\n",
      "Iteration 22: [2.3434112732891204, 2.2182041748301295, -0.15584296063012815]\n",
      "Iteration 23: [2.3435117838650643, 2.218365731516459, -0.1559639245898758]\n",
      "Iteration 24: [2.3435824140265837, 2.2184798776816055, -0.15604864937959634]\n",
      "Iteration 25: [2.3436321317653004, 2.2185599042619253, -0.15610843387795903]\n",
      "Iteration 26: [2.343667084534971, 2.2186163331737396, -0.1561503880578302]\n",
      "Iteration 27: [2.3436916803078924, 2.2186559535502104, -0.15617995017649006]\n",
      "Iteration 28: [2.3437089759316754, 2.2186838602202905, -0.15620071748166264]\n",
      "Iteration 29: [2.3437211444254884, 2.2187034702983337, -0.15621533927445128]\n",
      "Iteration 30: [2.3437297023931962, 2.218717274324797, -0.15622561700443108]\n",
      "Iteration 31: [2.343735722832307, 2.2187269788006865, -0.1562328502085581]\n",
      "Iteration 32: [2.343739957252311, 2.2187338077498078, -0.15623793608673608]\n",
      "Iteration 33: [2.3437429359591357, 2.2187386098085944, -0.1562415145503852]\n",
      "Iteration 34: [2.3437450310897447, 2.2187419883533024, -0.15624403111526491]\n",
      "Iteration 35: [2.3437465048671418, 2.2187443644400915, -0.15624580155926998]\n",
      "Iteration 36: [2.34374754149984, 2.218746035995227, -0.15624704674946485]\n",
      "Iteration 37: [2.3437482706861728, 2.2187472116662565, -0.15624792269805887]\n",
      "Iteration 38: [2.343748783591079, 2.218748038694097, -0.15624853880373718]\n",
      "Iteration 39: [2.3437491443744585, 2.2187486203995177, -0.15624897219585457]\n",
      "Iteration 40: [2.343749398148843, 2.218749029588723, -0.15624927703469887]\n",
      "Iteration 41: [2.3437495766558554, 2.2187493174060804, -0.15624949146525785]\n",
      "Iteration 42: [2.3437497022178344, 2.218749519862124, -0.15624964229360322]\n",
      "Iteration 43: [2.343749790538932, 2.218749662268347, -0.15624974838841652]\n",
      "Iteration 44: [2.3437498526641907, 2.218749762438588, -0.15624982301512524]\n",
      "Iteration 45: [2.3437498963634282, 2.218749832898147, -0.15624987550827338]\n",
      "Iteration 46: [2.343749927101605, 2.2187498824599916, -0.15624991243194444]\n",
      "Iteration 47: [2.343749948722984, 2.2187499173218312, -0.15624993840431767]\n",
      "Iteration 48: [2.343749963931537, 2.218749941843873, -0.1562499566733294]\n",
      "Iteration 49: [2.3437499746293007, 2.2187499590927318, -0.1562499695238566]\n",
      "Iteration 50: [2.343749982154147, 2.2187499712256713, -0.15624997856295283]\n",
      "Iteration 51: [2.343749987447156, 2.2187499797600174, -0.15624998492109796]\n",
      "Iteration 52: [2.343749991170279, 2.218749985763117, -0.15624998939343815]\n",
      "Iteration 53: [2.343749993789139, 2.2187499899857186, -0.1562499925393027]\n",
      "Iteration 54: [2.3437499956312555, 2.2187499929559147, -0.1562499947521152]\n",
      "Iteration 55: [2.3437499969270075, 2.218749995045162, -0.15624999630861697]\n",
      "Iteration 56: [2.3437499978384446, 2.218749996514747, -0.15624999740346635]\n",
      "Iteration 57: [2.343749998479553, 2.2187499975484593, -0.15624999817358773]\n",
      "Iteration 58: [2.3437499989305115, 2.2187499982755763, -0.15624999871529432]\n",
      "Converged in 58 iterations\n",
      "Final Solution: [2.3437499989305115, 2.2187499982755763, -0.15624999871529432]\n",
      "Number of iterations: 58\n",
      "Time taken (seconds): 0.023999929428100586\n"
     ]
    }
   ],
   "source": [
    "# Hard-coded Example Input (3x3 Matrix)\n",
    "A_3x3 = [4.0  -1.0   1.0;\n",
    "         -1.0  3.0  2.0;\n",
    "         1.0  2.0   5.0]\n",
    "\n",
    "b_3x3 = [7.0, 4.0, 6.0]  # Vector b\n",
    "\n",
    "x0_3x3 = [0, 0, 0]  # Initial guess, a vector of zeros (3 elements)\n",
    "\n",
    "\n",
    "function main()\n",
    "    # Start the timer\n",
    "    start_time = time()\n",
    "\n",
    "    # Call the Jacobi method with the matrix A_3x3, vector b_3x3, and initial guess x0_3x3\n",
    "    x_sol, iterations = jacobi(A_3x3, b_3x3, x0_3x3)\n",
    "\n",
    "    # End the timer\n",
    "    end_time = time()\n",
    "\n",
    "    # Calculate the duration\n",
    "    duration = end_time - start_time\n",
    "\n",
    "    # Output the final solution, number of iterations, and time taken\n",
    "    println(\"Final Solution: \", x_sol)\n",
    "    println(\"Number of iterations: \", iterations)\n",
    "    println(\"Time taken (seconds): \", duration)\n",
    "end\n",
    "\n",
    "# Call the main function\n",
    "main()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b56cd8dd-d37b-4d3e-8674-cc97eab0f25b",
   "metadata": {},
   "source": [
    "# Hard-coded Example Input (4x4 Matrix)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "efca2f07-5fd4-4693-ae78-1ba874c04bb1",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0: [0.6, 2.272727272727273, -1.1, 1.875]\n",
      "Iteration 1: [1.0472727272727274, 1.7159090909090908, -0.8052272727272726, 0.8852272727272728]\n",
      "Iteration 2: [0.9326363636363636, 2.053305785123967, -1.0493409090909092, 1.1308806818181818]\n",
      "Iteration 3: [1.0151987603305785, 1.9536957644628101, -0.9681086260330579, 0.9738427169421487]\n",
      "Iteration 4: [0.9889913016528926, 2.0114147257700976, -1.0102859039256198, 1.021350510072314]\n",
      "Iteration 5: [1.0031986533621338, 1.9922412606827573, -0.9945217367463373, 0.994433739845511]\n",
      "Iteration 6: [0.9981284734175432, 2.00230688155266, -1.0019722306196, 1.0035943101506737]\n",
      "Iteration 7: [1.0006251342791859, 1.9986703011223563, -0.9990355755131752, 0.9988883905903024]\n",
      "Iteration 8: [0.9996741452148707, 2.0004476715450092, -1.0003691576845712, 1.0006191901399695]\n",
      "Iteration 9: [1.0001185986914152, 1.9997679470100354, -0.9998281428744763, 0.9997859784600501]\n",
      "Iteration 10: [0.9999424232758989, 2.0000847745851624, -1.0000683271912745, 1.0001085020119271]\n",
      "Iteration 11: [1.000022142896771, 1.999958962731713, -0.9999691569954707, 0.9999596686316548]\n",
      "Iteration 12: [0.9999897276722655, 2.0000158163642126, -1.0000125654430174, 1.0000192443511737]\n",
      "Iteration 13: [1.0000040947250248, 1.9999926753796116, -0.9999944394629144, 0.9999924981830431]\n",
      "Iteration 14: [0.999998155430544, 2.00000292370118, -1.0000023015887396, 1.0000034417997814]\n",
      "Iteration 15: [1.0000007526878658, 1.9999986844038597, -0.9999989945360127, 0.9999986159134651]\n",
      "Iteration 16: [0.9999996673475884, 2.0000005373101324, -1.0000004205058406, 1.000000619031551]\n",
      "Iteration 17: [1.0000001378321814, 1.9999997627042811, -0.9999998178353493, 0.9999997459454703]\n",
      "Iteration 18: [0.999999939837498, 2.00000009837822, -1.0000000767014612, 1.0000001117564759]\n",
      "Iteration 19: [1.0000000251781143, 1.9999999570787828, -0.9999999669540302, 0.9999999535204849]\n",
      "Iteration 20: [0.9999999890986843, 2.00000001796933, -1.0000000139756962, 1.0000000202262027]\n",
      "Iteration 21: [1.000000004592072, 1.9999999922222165, -0.9999999940001836, 0.9999999915145392]\n",
      "Iteration 22: [0.9999999980222583, 2.0000000032771155, -1.0000000025447389, 1.0000000036666459]\n",
      "Iteration 23: [1.0000000008366592, 1.9999999985888712, -0.9999999989100756, 0.9999999984529894]\n",
      "Iteration 24: [0.9999999996409024, 2.000000000597056, -1.0000000004631457, 1.000000000665414]\n",
      "Iteration 25: [1.0000000001523346, 1.999999999743774, -0.9999999998019333, 0.9999999997182107]\n",
      "Converged in 25 iterations\n",
      "Final Solution: [1.0000000001523346, 1.999999999743774, -0.9999999998019333, 0.9999999997182107]\n",
      "Number of iterations: 25\n",
      "Time taken (seconds): 0.014000177383422852\n"
     ]
    }
   ],
   "source": [
    "# Hard-coded Example Input (4x4 Matrix)\n",
    "A_4x4 = [10.0  -1.0   2.0    0.0;\n",
    "         -1.0   11.0  -1.0   3.0;\n",
    "          2.0   -1.0  10.0  -1.0;\n",
    "          0.0    3.0  -1.0   8.0]\n",
    "\n",
    "b_4x4 = [6.0, 25.0, -11.0, 15.0]  # Vector b\n",
    "\n",
    "x0_4x4 = [0.0, 0.0, 0.0, 0.0]  # Initial guess, a vector of zeros (4 elements)\n",
    "\n",
    "function main()\n",
    "    # Start the timer\n",
    "    start_time = time()\n",
    "\n",
    "    # Call the Jacobi method with the matrix A_4x4, vector b_4x4, and initial guess x0_4x4\n",
    "    x_sol, iterations = jacobi(A_4x4, b_4x4, x0_4x4)\n",
    "\n",
    "    # End the timer\n",
    "    end_time = time()\n",
    "\n",
    "    # Calculate the duration\n",
    "    duration = end_time - start_time\n",
    "\n",
    "    # Output the final solution, number of iterations, and time taken\n",
    "    println(\"Final Solution: \", x_sol)\n",
    "    println(\"Number of iterations: \", iterations)\n",
    "    println(\"Time taken (seconds): \", duration)\n",
    "end\n",
    "\n",
    "# Call the main function\n",
    "main()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "81d00b21-5ce4-4bc6-aa47-bf90ffe8163d",
   "metadata": {},
   "source": [
    "# Hard-coded Example Input (5x5 Matrix)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "12d91156-1687-457b-b43e-a2c78f9bd90d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0: [0.6, 2.272727272727273, -1.1, 1.25, 2.0]\n",
      "Iteration 1: [1.0472727272727274, 1.8863636363636365, -0.8677272727272726, 0.7568181818181818, 2.25]\n",
      "Iteration 2: [0.9621818181818181, 2.0826446280991737, -1.0451363636363635, 0.8935984848484848, 2.1513636363636364]\n",
      "Iteration 3: [1.01729173553719, 2.0214772727272727, -0.9948120523415979, 0.8215244490358126, 2.178719696969697]\n",
      "Iteration 4: [1.001110137741047, 2.050718757826196, -1.0191581749311296, 0.8432896522038567, 2.1643048898071626]\n",
      "Iteration 5: [1.0089035107688455, 2.0410984551089406, -1.0108211865452041, 0.8327492034497871, 2.1686579304407716]\n",
      "Iteration 6: [1.0062740828199348, 2.045439519443116, -1.0143959362978963, 0.8362117815473954, 2.1665498406899575]\n",
      "Iteration 7: [1.0074231392038908, 2.0439311638072595, -1.013089686464936, 0.834652945505226, 2.167242356309479]\n",
      "Iteration 8: [1.0070110536737131, 2.0445795105657525, -1.0136262169095296, 0.8351965982018972, 2.166930589101045]\n",
      "Iteration 9: [1.0071831944384813, 2.04434500383259, -1.0134245998579776, 0.8349638200411881, 2.1670393196403794]\n",
      "Iteration 10: [1.0071194203548546, 2.0444424667688126, -1.0135057565003183, 0.8350483090237192, 2.1669927640082376]\n",
      "Iteration 11: [1.007145397976945, 2.044406248798489, -1.0134748064917178, 0.8350133006001235, 2.1670096618047436]\n",
      "Iteration 12: [1.0071355861781925, 2.0444209717895325, -1.0134871246555277, 0.8350263424097966, 2.167002660120025]\n",
      "Iteration 13: [1.0071395221100587, 2.0444154031175703, -1.0134823858157056, 0.8350210516746582, 2.1670052684819594]\n",
      "Iteration 14: [1.007138017474898, 2.0444176346609435, -1.0134842589427888, 0.8350230561094619, 2.1670042103349316]\n",
      "Iteration 15: [1.0071386152546522, 2.044416780927611, -1.0134835344179391, 0.835022253950776, 2.167004611221892]\n",
      "Iteration 16: [1.0071383849763489, 2.044417119907671, -1.0134838195630917, 0.8350225611684267, 2.167004450790155]\n",
      "Iteration 17: [1.0071384759033855, 2.0444169892643616, -1.01348370888766, 0.835022439292004, 2.167004512233685]\n",
      "Iteration 18: [1.0071384407039683, 2.0444170408308833, -1.0134837523250406, 0.8350224862960783, 2.1670044878584007]\n",
      "Iteration 19: [1.0071384545480964, 2.0444170208627903, -1.0134837354280974, 0.8350224677533925, 2.1670044972592155]\n",
      "Iteration 20: [1.0071384491718987, 2.044417028714529, -1.013483742048001, 0.8350224749368955, 2.1670044935506785]\n",
      "Iteration 21: [1.0071384512810533, 2.0444170256648375, -1.0134837394692373, 0.8350224721132574, 2.167004494987379]\n",
      "Iteration 22: [1.0071384504603311, 2.044417026861095, -1.0134837404784012, 0.8350224732103024, 2.1670044944226516]\n",
      "Iteration 23: [1.0071384507817895, 2.0444170263955477, -1.0134837400849264, 0.8350224727800805, 2.16700449464206]\n",
      "Converged in 23 iterations\n",
      "Final Solution: [1.0071384507817895, 2.0444170263955477, -1.0134837400849264, 0.8350224727800805, 2.16700449464206]\n",
      "Number of iterations: 23\n",
      "Time taken (seconds): 0.014999866485595703\n"
     ]
    }
   ],
   "source": [
    "# Hard-coded Example Input (5x5 Matrix)\n",
    "A_5x5 = [10.0  -1.0   2.0   0.0   0.0;\n",
    "         -1.0  11.0  -1.0   3.0   0.0;\n",
    "          2.0  -1.0  10.0  -1.0   0.0;\n",
    "          0.0   3.0  -1.0  12.0  -1.0;\n",
    "          0.0   0.0   0.0  -1.0   5.0]\n",
    "\n",
    "b_5x5 = [6.0, 25.0, -11.0, 15.0, 10.0]  # Vector b\n",
    "\n",
    "x0_5x5 = zeros(5)  # Initial guess, a vector of zeros (5 elements)\n",
    "\n",
    "function main()\n",
    "    # Start the timer\n",
    "    start_time = time()\n",
    "\n",
    "    # Call the Jacobi method with the matrix A_5x5, vector b_5x5, and initial guess x0_5x5\n",
    "    x_sol, iterations = jacobi(A_5x5, b_5x5, x0_5x5)\n",
    "\n",
    "    # End the timer\n",
    "    end_time = time()\n",
    "\n",
    "    # Calculate the duration\n",
    "    duration = end_time - start_time\n",
    "\n",
    "    # Output the final solution, number of iterations, and time taken\n",
    "    println(\"Final Solution: \", x_sol)\n",
    "    println(\"Number of iterations: \", iterations)\n",
    "    println(\"Time taken (seconds): \", duration)\n",
    "end\n",
    "\n",
    "# Call the main function\n",
    "main()\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6e68c686-b4a4-4447-a41b-b2f982bf34e4",
   "metadata": {},
   "source": [
    "# Hard-coded Example Input (6x6 Matrix)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "af2063ba-99c6-46be-b95a-39cf540f6e13",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0: [1.0, 0.25, 0.2, 1.0, 1.5, 0.5]\n",
      "Iteration 1: [1.0625, 0.55, 0.325, 1.425, 1.875, 0.875]\n",
      "Iteration 2: [1.1375, 0.596875, 0.3975, 1.55, 2.075, 0.96875]\n",
      "Iteration 3: [1.14921875, 0.63375, 0.4146875, 1.618125, 2.1296875, 1.01875]\n",
      "Iteration 4: [1.1584375, 0.6409765625, 0.4251875, 1.6360937500000001, 2.15921875, 1.032421875]\n",
      "Iteration 5: [1.160244140625, 0.64590625, 0.42770703125000004, 1.6461015625, 2.1671289062500003, 1.0398046875]\n",
      "Iteration 6: [1.1614765625, 0.64698779296875, 0.42920078124999994, 1.6487089843750002, 2.1714765625, 1.0417822265625]\n",
      "Iteration 7: [1.1617469482421874, 0.6476693359375, 0.429569677734375, 1.6501693359375, 2.172622802734375, 1.0428691406250001]\n",
      "Iteration 8: [1.161917333984375, 0.6478291564941405, 0.42978386718750006, 1.6505481201171874, 2.173259619140625, 1.0431557006835939]\n",
      "Iteration 9: [1.161957289123535, 0.6479253002929688, 0.4298377276611328, 1.6507608715820312, 2.1734259552001953, 1.0433149047851562]\n",
      "Iteration 10: [1.1619813250732423, 0.6479487541961669, 0.42986861718749997, 1.6508159207153321, 2.173518944091797, 1.0433564888000488]\n",
      "Iteration 11: [1.1619871885490418, 0.6479624855651855, 0.42987646749114994, 1.6508468903198241, 2.1735431023788454, 1.043379736022949]\n",
      "Iteration 12: [1.1619906213912965, 0.647965914010048, 0.429880937588501, 1.6508548924674988, 2.1735566565856934, 1.0433857755947114]\n",
      "Iteration 13: [1.161991478502512, 0.6479678897449493, 0.4298820806477547, 1.6508593985435485, 2.1735601670155527, 1.0433891641464235]\n",
      "Iteration 14: [1.1619919724362373, 0.6479683897875667, 0.4298827288288498, 1.650860561915827, 2.1735621406724928, 1.0433900417538882]\n",
      "Iteration 15: [1.1619920974468916, 0.6479686753162718, 0.4298828951703394, 1.6508612173753356, 2.173562650917429, 1.0433905351681232]\n",
      "Iteration 16: [1.161992168829068, 0.6479687481543077, 0.42988298926916074, 1.650861386521942, 2.1735629381358645, 1.0433906627293572]\n",
      "Iteration 17: [1.161992187038577, 0.6479687895245572, 0.4298830134676249, 1.6508614818512564, 2.1735630123128247, 1.0433907345339661]\n",
      "Iteration 18: [1.1619921973811393, 0.6479688001265504, 0.4298830271375813, 1.6508615064451124, 2.173563054096306, 1.043390753078206]\n",
      "Iteration 19: [1.1619922000316376, 0.6479688061296802, 0.42988303065716627, 1.6508615203084718, 2.1735630648808297, 1.0433907635240764]\n",
      "Iteration 20: [1.16199220153242, 0.647968807672201, 0.42988303264381517, 1.650861523884499, 2.173563070958137, 1.0433907662202073]\n",
      "Iteration 21: [1.1619922019180502, 0.6479688085440588, 0.42988303315567, 1.650861525900488, 2.1735630725261768, 1.0433907677395342]\n",
      "Iteration 22: [1.1619922021360147, 0.6479688087684301, 0.4298830334444547, 1.6508615264204618, 2.1735630734100058, 1.0433907681315442]\n",
      "Converged in 22 iterations\n",
      "Iteration 0: [1.0, 0.25, 0.2, 1.0, 1.5, 0.5]\n",
      "Iteration 1: [1.0625, 0.55, 0.325, 1.425, 1.875, 0.875]\n",
      "Iteration 2: [1.1375, 0.596875, 0.3975, 1.55, 2.075, 0.96875]\n",
      "Iteration 3: [1.14921875, 0.63375, 0.4146875, 1.618125, 2.1296875, 1.01875]\n",
      "Iteration 4: [1.1584375, 0.6409765625, 0.4251875, 1.6360937500000001, 2.15921875, 1.032421875]\n",
      "Iteration 5: [1.160244140625, 0.64590625, 0.42770703125000004, 1.6461015625, 2.1671289062500003, 1.0398046875]\n",
      "Iteration 6: [1.1614765625, 0.64698779296875, 0.42920078124999994, 1.6487089843750002, 2.1714765625, 1.0417822265625]\n",
      "Iteration 7: [1.1617469482421874, 0.6476693359375, 0.429569677734375, 1.6501693359375, 2.172622802734375, 1.0428691406250001]\n",
      "Iteration 8: [1.161917333984375, 0.6478291564941405, 0.42978386718750006, 1.6505481201171874, 2.173259619140625, 1.0431557006835939]\n",
      "Iteration 9: [1.161957289123535, 0.6479253002929688, 0.4298377276611328, 1.6507608715820312, 2.1734259552001953, 1.0433149047851562]\n",
      "Iteration 10: [1.1619813250732423, 0.6479487541961669, 0.42986861718749997, 1.6508159207153321, 2.173518944091797, 1.0433564888000488]\n",
      "Iteration 11: [1.1619871885490418, 0.6479624855651855, 0.42987646749114994, 1.6508468903198241, 2.1735431023788454, 1.043379736022949]\n",
      "Iteration 12: [1.1619906213912965, 0.647965914010048, 0.429880937588501, 1.6508548924674988, 2.1735566565856934, 1.0433857755947114]\n",
      "Iteration 13: [1.161991478502512, 0.6479678897449493, 0.4298820806477547, 1.6508593985435485, 2.1735601670155527, 1.0433891641464235]\n",
      "Iteration 14: [1.1619919724362373, 0.6479683897875667, 0.4298827288288498, 1.650860561915827, 2.1735621406724928, 1.0433900417538882]\n",
      "Iteration 15: [1.1619920974468916, 0.6479686753162718, 0.4298828951703394, 1.6508612173753356, 2.173562650917429, 1.0433905351681232]\n",
      "Iteration 16: [1.161992168829068, 0.6479687481543077, 0.42988298926916074, 1.650861386521942, 2.1735629381358645, 1.0433906627293572]\n",
      "Iteration 17: [1.161992187038577, 0.6479687895245572, 0.4298830134676249, 1.6508614818512564, 2.1735630123128247, 1.0433907345339661]\n",
      "Iteration 18: [1.1619921973811393, 0.6479688001265504, 0.4298830271375813, 1.6508615064451124, 2.173563054096306, 1.043390753078206]\n",
      "Iteration 19: [1.1619922000316376, 0.6479688061296802, 0.42988303065716627, 1.6508615203084718, 2.1735630648808297, 1.0433907635240764]\n",
      "Iteration 20: [1.16199220153242, 0.647968807672201, 0.42988303264381517, 1.650861523884499, 2.173563070958137, 1.0433907662202073]\n",
      "Iteration 21: [1.1619922019180502, 0.6479688085440588, 0.42988303315567, 1.650861525900488, 2.1735630725261768, 1.0433907677395342]\n",
      "Iteration 22: [1.1619922021360147, 0.6479688087684301, 0.4298830334444547, 1.6508615264204618, 2.1735630734100058, 1.0433907681315442]\n",
      "Converged in 22 iterations\n",
      "Final Solution: [1.1619922021360147, 0.6479688087684301, 0.4298830334444547, 1.6508615264204618, 2.1735630734100058, 1.0433907681315442]\n",
      "Number of iterations: 22\n",
      "Time taken (seconds): 0.015999794006347656\n"
     ]
    }
   ],
   "source": [
    "# Hard-coded Example Input (6x6 Matrix)\n",
    "A_6x6 = [4.0  -1.0   0.0    0.0    0.0   0.0;\n",
    "         -1.0  4.0  -1.0    0.0    0.0   0.0;\n",
    "          0.0  -1.0  10.0   -1.0    0.0   0.0;\n",
    "          0.0   0.0  -1.0   4.0   -1.0   0.0;\n",
    "          0.0   0.0   0.0   -1.0    4.0  -1.0;\n",
    "          0.0   0.0   0.0    0.0   -1.0  4.0]\n",
    "\n",
    "b_6x6 = [4.0, 1.0, 2.0, 4.0, 6.0, 2.0]  # Vector b\n",
    "\n",
    "x0_6x6 = zeros(6)  # Initial guess, a vector of zeros (6 elements)\n",
    "\n",
    "# Call the Jacobi method for the 6x6 matrix\n",
    "x_sol_6x6, iterations_6x6 = jacobi(A_6x6, b_6x6, x0_6x6)\n",
    "\n",
    "function main()\n",
    "    # Start the timer\n",
    "    start_time = time()\n",
    "\n",
    "    # Call the Jacobi method with the matrix A_6x6, vector b_6x6, and initial guess x0_6x6\n",
    "    x_sol, iterations = jacobi(A_6x6, b_6x6, x0_6x6)\n",
    "\n",
    "    # End the timer\n",
    "    end_time = time()\n",
    "\n",
    "    # Calculate the duration\n",
    "    duration = end_time - start_time\n",
    "\n",
    "    # Output the final solution, number of iterations, and time taken\n",
    "    println(\"Final Solution: \", x_sol)\n",
    "    println(\"Number of iterations: \", iterations)\n",
    "    println(\"Time taken (seconds): \", duration)\n",
    "end\n",
    "\n",
    "# Call the main function\n",
    "main()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a01208cc-f4fb-4d56-a69a-e096406c318b",
   "metadata": {},
   "source": [
    "# Hard-coded Example Input (7x7 Matrix)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "3729f497-c360-41bc-b298-9faeb52f8d01",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0: [0.6, 2.272727272727273, -1.1, 1.25, -0.8888888888888888, 1.0, 0.7142857142857143]\n",
      "Iteration 1: [1.0472727272727274, 1.8863636363636365, -1.1677272727272725, 0.516077441077441, -0.5277777777777778, 1.2236507936507937, 0.8571428571428571]\n",
      "Iteration 2: [1.0221818181818183, 2.1210284664830117, -1.4363056758056758, 0.6371170033670034, -0.5596245524023301, 1.330476911976912, 0.889092970521542]\n",
      "Iteration 3: [1.0993639818094363, 2.0613204665704665, -1.4277648902444358, 0.5534153643619133, -0.5224365747421302, 1.407876089313391, 0.9043538445681303]\n",
      "Iteration 4: [1.0916850247059338, 2.0919411816799327, -1.4807620400626667, 0.5721530946085028, -0.5145369396679228, 1.4142775365817175, 0.915410869901913]\n",
      "Iteration 5: [1.1053465261805266, 2.081314881892524, -1.4762108382868584, 0.5607397896024677, -0.5110324258031179, 1.4328623110754068, 0.9163253623688169]\n",
      "Iteration 6: [1.103373655846624, 2.086083301735115, -1.4867225314092283, 0.5640676741860378, -0.508170620916302, 1.4322893025623156, 0.9189803301536296]\n",
      "Iteration 7: [1.1059528364553572, 2.08404073653448, -1.4853464243459042, 0.5622380785390938, -0.5079281911877034, 1.436280668254871, 0.9188984717946165]\n",
      "Iteration 8: [1.1054733585226288, 2.0848992887720157, -1.4874468862601753, 0.562883597905246, -0.5072445094390182, 1.4359081362456922, 0.919468666893553]\n",
      "Iteration 9: [1.1059793061292367, 2.0844886980497015, -1.4870888239105073, 0.5625508948320633, -0.5072555699559299, 1.4367320306796043, 0.9194154480350989]\n",
      "Iteration 10: [1.1058666345870716, 2.08465798161114, -1.4875115111415522, 0.5626824593320382, -0.5071094493120809, 1.436617077985476, 0.9195331472399434]\n",
      "Iteration 11: [1.1059681003894244, 2.0845734314044915, -1.4874244062187394, 0.562617091226079, -0.5071203760774455, 1.436784878204044, 0.9195167254264965]\n",
      "Iteration 12: [1.105942224384197, 2.08460840186295, -1.487510031276041, 0.5626445769575285, -0.5070903502628704, 1.4367549191927824, 0.919540696886292]\n",
      "Iteration 13: [1.105962846441503, 2.0845907692941426, -1.4874896227526262, 0.5626312010726866, -0.5070939538507674, 1.4367890090188673, 0.9195364170275404]\n",
      "Iteration 14: [1.1059570014799394, 2.084598147315529, -1.4875070749572779, 0.5626370096261816, -0.5070878645432866, 1.4367817377583882, 0.9195412870026953]\n",
      "Iteration 15: [1.1059612297230086, 2.084594445240374, -1.4875024059293334, 0.5626342182127374, -0.5070888349841157, 1.4367886782787955, 0.9195402482511984]\n",
      "Iteration 16: [1.1059599257099042, 2.084596015377769, -1.4875059830829291, 0.5626354519471191, -0.5070876028032969, 1.4367869796070967, 0.9195412397541137]\n",
      "Iteration 17: [1.1059607981543629, 2.084595235162329, -1.487504932291621, 0.5626348639983721, -0.5070878432042986, 1.4367883983396308, 0.9195409970867281]\n",
      "Iteration 18: [1.105960509974557, 2.084595570351602, -1.487505669216692, 0.5626351265847579, -0.5070875932580408, 1.4367880107552993, 0.9195411997628044]\n",
      "Iteration 19: [1.1059606908784985, 2.084595405545781, -1.4875054355278654, 0.5626350022058718, -0.507087650211627, 1.43678830208968, 0.9195411443936141]\n",
      "Iteration 20: [1.1059606276601512, 2.084595477157547, -1.4875055880274384, 0.5626350581352636, -0.5070875992905298, 1.4367882150553957, 0.9195411860128113]\n",
      "Iteration 21: [1.1059606653212424, 2.0845954422933564, -1.487505536519368, 0.5626350317674492, -0.507087612417105, 1.4367882751514067, 0.9195411735793423]\n",
      "Iteration 22: [1.1059606515332092, 2.0845954575908663, -1.48750556820359, 0.5626350436819547, -0.507087601992193, 1.4367882558303235, 0.9195411821644867]\n",
      "Iteration 23: [1.1059606593998046, 2.084595450207614, -1.4875055569284568, 0.5626350380859683, -0.5070876049619331, 1.4367882682790871, 0.919541179404332]\n",
      "Iteration 24: [1.1059606564064528, 2.0845954534739493, -1.4875055635343288, 0.5626350406238974, -0.5070876028173176, 1.4367882640265837, 0.9195411811827267]\n",
      "Iteration 25: [1.1059606580542607, 2.0845954519091303, -1.487505561079481, 0.5626350394355422, -0.5070876034803261, 1.4367882666151077, 0.9195411805752263]\n",
      "Iteration 26: [1.105960657406809, 2.084595452606196, -1.487505562460917, 0.5626350399760668, -0.5070876030371381, 1.4367882656853017, 0.9195411809450154]\n",
      "Iteration 27: [1.105960657752803, 2.0845954522743355, -1.487505561928726, 0.5626350397236131, -0.5070876031837034, 1.4367882662253488, 0.9195411808121859]\n",
      "Converged in 27 iterations\n",
      "Iteration 0: [0.6, 2.272727272727273, -1.1, 1.25, -0.8888888888888888, 1.0, 0.7142857142857143]\n",
      "Iteration 1: [1.0472727272727274, 1.8863636363636365, -1.1677272727272725, 0.516077441077441, -0.5277777777777778, 1.2236507936507937, 0.8571428571428571]\n",
      "Iteration 2: [1.0221818181818183, 2.1210284664830117, -1.4363056758056758, 0.6371170033670034, -0.5596245524023301, 1.330476911976912, 0.889092970521542]\n",
      "Iteration 3: [1.0993639818094363, 2.0613204665704665, -1.4277648902444358, 0.5534153643619133, -0.5224365747421302, 1.407876089313391, 0.9043538445681303]\n",
      "Iteration 4: [1.0916850247059338, 2.0919411816799327, -1.4807620400626667, 0.5721530946085028, -0.5145369396679228, 1.4142775365817175, 0.915410869901913]\n",
      "Iteration 5: [1.1053465261805266, 2.081314881892524, -1.4762108382868584, 0.5607397896024677, -0.5110324258031179, 1.4328623110754068, 0.9163253623688169]\n",
      "Iteration 6: [1.103373655846624, 2.086083301735115, -1.4867225314092283, 0.5640676741860378, -0.508170620916302, 1.4322893025623156, 0.9189803301536296]\n",
      "Iteration 7: [1.1059528364553572, 2.08404073653448, -1.4853464243459042, 0.5622380785390938, -0.5079281911877034, 1.436280668254871, 0.9188984717946165]\n",
      "Iteration 8: [1.1054733585226288, 2.0848992887720157, -1.4874468862601753, 0.562883597905246, -0.5072445094390182, 1.4359081362456922, 0.919468666893553]\n",
      "Iteration 9: [1.1059793061292367, 2.0844886980497015, -1.4870888239105073, 0.5625508948320633, -0.5072555699559299, 1.4367320306796043, 0.9194154480350989]\n",
      "Iteration 10: [1.1058666345870716, 2.08465798161114, -1.4875115111415522, 0.5626824593320382, -0.5071094493120809, 1.436617077985476, 0.9195331472399434]\n",
      "Iteration 11: [1.1059681003894244, 2.0845734314044915, -1.4874244062187394, 0.562617091226079, -0.5071203760774455, 1.436784878204044, 0.9195167254264965]\n",
      "Iteration 12: [1.105942224384197, 2.08460840186295, -1.487510031276041, 0.5626445769575285, -0.5070903502628704, 1.4367549191927824, 0.919540696886292]\n",
      "Iteration 13: [1.105962846441503, 2.0845907692941426, -1.4874896227526262, 0.5626312010726866, -0.5070939538507674, 1.4367890090188673, 0.9195364170275404]\n",
      "Iteration 14: [1.1059570014799394, 2.084598147315529, -1.4875070749572779, 0.5626370096261816, -0.5070878645432866, 1.4367817377583882, 0.9195412870026953]\n",
      "Iteration 15: [1.1059612297230086, 2.084594445240374, -1.4875024059293334, 0.5626342182127374, -0.5070888349841157, 1.4367886782787955, 0.9195402482511984]\n",
      "Iteration 16: [1.1059599257099042, 2.084596015377769, -1.4875059830829291, 0.5626354519471191, -0.5070876028032969, 1.4367869796070967, 0.9195412397541137]\n",
      "Iteration 17: [1.1059607981543629, 2.084595235162329, -1.487504932291621, 0.5626348639983721, -0.5070878432042986, 1.4367883983396308, 0.9195409970867281]\n",
      "Iteration 18: [1.105960509974557, 2.084595570351602, -1.487505669216692, 0.5626351265847579, -0.5070875932580408, 1.4367880107552993, 0.9195411997628044]\n",
      "Iteration 19: [1.1059606908784985, 2.084595405545781, -1.4875054355278654, 0.5626350022058718, -0.507087650211627, 1.43678830208968, 0.9195411443936141]\n",
      "Iteration 20: [1.1059606276601512, 2.084595477157547, -1.4875055880274384, 0.5626350581352636, -0.5070875992905298, 1.4367882150553957, 0.9195411860128113]\n",
      "Iteration 21: [1.1059606653212424, 2.0845954422933564, -1.487505536519368, 0.5626350317674492, -0.507087612417105, 1.4367882751514067, 0.9195411735793423]\n",
      "Iteration 22: [1.1059606515332092, 2.0845954575908663, -1.48750556820359, 0.5626350436819547, -0.507087601992193, 1.4367882558303235, 0.9195411821644867]\n",
      "Iteration 23: [1.1059606593998046, 2.084595450207614, -1.4875055569284568, 0.5626350380859683, -0.5070876049619331, 1.4367882682790871, 0.919541179404332]\n",
      "Iteration 24: [1.1059606564064528, 2.0845954534739493, -1.4875055635343288, 0.5626350406238974, -0.5070876028173176, 1.4367882640265837, 0.9195411811827267]\n",
      "Iteration 25: [1.1059606580542607, 2.0845954519091303, -1.487505561079481, 0.5626350394355422, -0.5070876034803261, 1.4367882666151077, 0.9195411805752263]\n",
      "Iteration 26: [1.105960657406809, 2.084595452606196, -1.487505562460917, 0.5626350399760668, -0.5070876030371381, 1.4367882656853017, 0.9195411809450154]\n",
      "Iteration 27: [1.105960657752803, 2.0845954522743355, -1.487505561928726, 0.5626350397236131, -0.5070876031837034, 1.4367882662253488, 0.9195411808121859]\n",
      "Converged in 27 iterations\n",
      "Final Solution: [1.105960657752803, 2.0845954522743355, -1.487505561928726, 0.5626350397236131, -0.5070876031837034, 1.4367882662253488, 0.9195411808121859]\n",
      "Number of iterations: 27\n",
      "Time taken (seconds): 0.021999835968017578\n"
     ]
    }
   ],
   "source": [
    "# Hard-coded Example Input (7x7 Matrix)\n",
    "A_7x7 = [10.0  -1.0   2.0    0.0    0.0   0.0   0.0;\n",
    "         -1.0  11.0  -1.0    3.0    0.0   0.0   0.0;\n",
    "          2.0  -1.0  10.0   -1.0    0.0   3.0   0.0;\n",
    "          0.0   3.0  -1.0   12.0   -1.0   0.0   0.0;\n",
    "          0.0   0.0   0.0   -1.0    9.0  -2.0   0.0;\n",
    "          0.0   0.0   3.0    0.0   -2.0  10.0  -1.0;\n",
    "          0.0   0.0   0.0    0.0    0.0  -1.0  7.0]\n",
    "\n",
    "b_7x7 = [6.0, 25.0, -11.0, 15.0, -8.0, 10.0, 5.0]  # Vector b\n",
    "\n",
    "x0_7x7 = zeros(7)  # Initial guess, a vector of zeros (7 elements)\n",
    "\n",
    "# Call the Jacobi method for the 7x7 matrix\n",
    "x_sol_7x7, iterations_7x7 = jacobi(A_7x7, b_7x7, x0_7x7)\n",
    "\n",
    "function main()\n",
    "    # Start the timer\n",
    "    start_time = time()\n",
    "\n",
    "    # Call the Jacobi method with the matrix A_7x7, vector b_7x7, and initial guess x0_7x7\n",
    "    x_sol, iterations = jacobi(A_7x7, b_7x7, x0_7x7)\n",
    "\n",
    "    # End the timer\n",
    "    end_time = time()\n",
    "\n",
    "    # Calculate the duration\n",
    "    duration = end_time - start_time\n",
    "\n",
    "    # Output the final solution, number of iterations, and time taken\n",
    "    println(\"Final Solution: \", x_sol)\n",
    "    println(\"Number of iterations: \", iterations)\n",
    "    println(\"Time taken (seconds): \", duration)\n",
    "end\n",
    "\n",
    "# Call the main function\n",
    "main()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "65e65e18-d311-427d-bafb-e0bce525fab4",
   "metadata": {},
   "source": [
    "# Hard-coded Example Input (8x8 Matrix)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "8a215836-79f7-4055-b629-f4a299a12fbe",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0: [0.6, 2.272727272727273, -1.1, 1.25, -0.8888888888888888, 1.0, 0.5, 1.5]\n",
      "Iteration 1: [1.0472727272727274, 1.8863636363636365, -1.1677272727272725, 0.516077441077441, -0.5277777777777778, 1.2022222222222223, 0.75, 1.5625]\n",
      "Iteration 2: [1.0221818181818183, 2.1210284664830117, -1.4298771043771044, 0.6371170033670034, -0.5643864571642349, 1.319762626262626, 0.7764722222222222, 1.59375]\n",
      "Iteration 3: [1.098078267523722, 2.061904882154882, -1.42455060453015, 0.5535542532508021, -0.5248175271230827, 1.3937330621025066, 0.7913512626262627, 1.5970590277777779]\n",
      "Iteration 4: [1.091100609121518, 2.092078627567379, -1.4761896585949281, 0.5720764351568434, -0.5176644025049094, 1.4015368021970547, 0.7990792089880284, 1.5989189078282828]\n",
      "Iteration 5: [1.1044457944757236, 2.0816983313687327, -1.472265656210998, 0.5608258380165022, -0.513872217827672, 1.4192319379762997, 0.8000455710025338, 1.5998849011235035]\n",
      "Iteration 6: [1.1026229643790728, 2.0863366022013836, -1.4824063233495113, 0.5640639276545943, -0.5111900317812109, 1.4189098103980184, 0.8019116839099804, 1.6000056963753166]\n",
      "Iteration 7: [1.1051149248900407, 2.084365896187798, -1.4811574830096224, 0.5622828198554272, -0.5109018279499299, 1.4226750590396091, 0.8018915506773336, 1.6002389604887475]\n",
      "Iteration 8: [1.1046680862207043, 2.0851917256649215, -1.4831606310855683, 0.5629035833730877, -0.5102630068961506, 1.422356034380634, 0.8022914019528358, 1.6002364438346668]\n",
      "Iteration 9: [1.1051512987836059, 2.0847997004559886, -1.4828308966545303, 0.562583432085293, -0.5102649275406272, 1.4231247281417239, 0.8022592478215301, 1.6002864252441045]\n",
      "Iteration 10: [1.105046149376505, 2.084960918715745, -1.48322936494511, 0.5627087562030731, -0.51012967907014, 1.4230222082703867, 0.8023411153385828, 1.6002824059776912]\n",
      "Iteration 11: [1.1051419648605965, 2.084880955983834, -1.483148924864535, 0.5626465166531261, -0.5101385363617948, 1.4231769852033633, 0.8023304614248078, 1.600292639417323]\n",
      "Iteration 12: [1.1051178805712905, 2.0849139536396986, -1.4832287412694325, 0.562672472568514, -0.5101110569933497, 1.4231500163294823, 0.8023469624620686, 1.600291307678101]\n",
      "Iteration 13: [1.1051371436178563, 2.0848974292360287, -1.4832099383922814, 0.5626598617348436, -0.5101141660858357, 1.4231811072283667, 0.8023441324007583, 1.6002933703077586]\n",
      "Iteration 14: [1.1051317306020592, 2.084904329092822, -1.4832260317949941, 0.5626653006511497, -0.5101086582009359, 1.423174561540593, 0.8023474477536127, 1.6002930165500948]\n",
      "Iteration 15: [1.1051356392682812, 2.084900890623056, -1.4832217516081925, 0.5626626935604669, -0.5101095084741849, 1.4231808226736724, 0.8023467578090688, 1.6002934309692016]\n",
      "Iteration 16: [1.1051344393839442, 2.0849023460889717, -1.4832250162374057, 0.5626638390040378, -0.5101084067880209, 1.4231792995685275, 0.8023474253642874, 1.6002933447261336]\n",
      "Iteration 17: [1.1051352378563783, 2.0849016278304022, -1.4832240592380461, 0.5626632948923049, -0.5101086179843229, 1.4231805660500463, 0.802347264429466, 1.6002934281705359]\n",
      "Iteration 18: [1.1051349746306494, 2.084901935812856, -1.483224725114019, 0.562663536607202, -0.5101083970008448, 1.4231802206174957, 0.8023473994220582, 1.6002934080536833]\n",
      "Iteration 19: [1.1051351386040893, 2.0849017854268204, -1.4832245138693727, 0.5626634225372139, -0.510108446906423, 1.4231804780762425, 0.802347362867118, 1.6002934249277572]\n",
      "Iteration 20: [1.1051350813165566, 2.0849018506475523, -1.4832246503472872, 0.5626634735786453, -0.5101084023678113, 1.423180401066239, 0.8023473903003999, 1.6002934203583898]\n",
      "Iteration 21: [1.1051351151342126, 2.084901819112121, -1.4832246041605632, 0.5626634496118538, -0.5101084138098751, 1.4231804536606638, 0.8023473821424629, 1.60029342378755]\n",
      "Iteration 22: [1.1051351027433247, 2.0849018329216444, -1.4832246322526441, 0.5626634603910997, -0.510108404785202, 1.4231804367004401, 0.8023473877448215, 1.6002934227678078]\n",
      "Iteration 23: [1.1051351097426934, 2.0849018263015804, -1.4832246222275227, 0.5626634553497684, -0.5101084073564467, 1.423180447493235, 0.8023473859468246, 1.6002934234681028]\n",
      "Iteration 24: [1.1051351070756625, 2.0849018292241697, -1.4832246280313743, 0.5626634576259408, -0.5101084055181957, 1.42318044379165, 0.8023473870961337, 1.600293423243353]\n",
      "Iteration 25: [1.1051351085286918, 2.084901827833315, -1.4832246258676165, 0.5626634565648269, -0.5101084060878621, 1.4231804460153865, 0.8023473867035003, 1.6002934233870167]\n",
      "Iteration 26: [1.1051351079568548, 2.0849018284515086, -1.48322462707054, 0.5626634570453813, -0.5101084057116, 1.4231804452130625, 0.8023473869402403, 1.6002934233379376]\n",
      "Iteration 27: [1.1051351082592589, 2.084901828159106, -1.4832246266056006, 0.5626634568219445, -0.5101084058364993, 1.423180445672866, 0.8023473868551001, 1.60029342336753]\n",
      "Converged in 27 iterations\n",
      "8x8 Final Solution: [1.1051351082592589, 2.084901828159106, -1.4832246266056006, 0.5626634568219445, -0.5101084058364993, 1.423180445672866, 0.8023473868551001, 1.60029342336753]\n",
      "8x8 Number of iterations: 27\n",
      "Iteration 0: [0.6, 2.272727272727273, -1.1, 1.25, -0.8888888888888888, 1.0, 0.5, 1.5]\n",
      "Iteration 1: [1.0472727272727274, 1.8863636363636365, -1.1677272727272725, 0.516077441077441, -0.5277777777777778, 1.2022222222222223, 0.75, 1.5625]\n",
      "Iteration 2: [1.0221818181818183, 2.1210284664830117, -1.4298771043771044, 0.6371170033670034, -0.5643864571642349, 1.319762626262626, 0.7764722222222222, 1.59375]\n",
      "Iteration 3: [1.098078267523722, 2.061904882154882, -1.42455060453015, 0.5535542532508021, -0.5248175271230827, 1.3937330621025066, 0.7913512626262627, 1.5970590277777779]\n",
      "Iteration 4: [1.091100609121518, 2.092078627567379, -1.4761896585949281, 0.5720764351568434, -0.5176644025049094, 1.4015368021970547, 0.7990792089880284, 1.5989189078282828]\n",
      "Iteration 5: [1.1044457944757236, 2.0816983313687327, -1.472265656210998, 0.5608258380165022, -0.513872217827672, 1.4192319379762997, 0.8000455710025338, 1.5998849011235035]\n",
      "Iteration 6: [1.1026229643790728, 2.0863366022013836, -1.4824063233495113, 0.5640639276545943, -0.5111900317812109, 1.4189098103980184, 0.8019116839099804, 1.6000056963753166]\n",
      "Iteration 7: [1.1051149248900407, 2.084365896187798, -1.4811574830096224, 0.5622828198554272, -0.5109018279499299, 1.4226750590396091, 0.8018915506773336, 1.6002389604887475]\n",
      "Iteration 8: [1.1046680862207043, 2.0851917256649215, -1.4831606310855683, 0.5629035833730877, -0.5102630068961506, 1.422356034380634, 0.8022914019528358, 1.6002364438346668]\n",
      "Iteration 9: [1.1051512987836059, 2.0847997004559886, -1.4828308966545303, 0.562583432085293, -0.5102649275406272, 1.4231247281417239, 0.8022592478215301, 1.6002864252441045]\n",
      "Iteration 10: [1.105046149376505, 2.084960918715745, -1.48322936494511, 0.5627087562030731, -0.51012967907014, 1.4230222082703867, 0.8023411153385828, 1.6002824059776912]\n",
      "Iteration 11: [1.1051419648605965, 2.084880955983834, -1.483148924864535, 0.5626465166531261, -0.5101385363617948, 1.4231769852033633, 0.8023304614248078, 1.600292639417323]\n",
      "Iteration 12: [1.1051178805712905, 2.0849139536396986, -1.4832287412694325, 0.562672472568514, -0.5101110569933497, 1.4231500163294823, 0.8023469624620686, 1.600291307678101]\n",
      "Iteration 13: [1.1051371436178563, 2.0848974292360287, -1.4832099383922814, 0.5626598617348436, -0.5101141660858357, 1.4231811072283667, 0.8023441324007583, 1.6002933703077586]\n",
      "Iteration 14: [1.1051317306020592, 2.084904329092822, -1.4832260317949941, 0.5626653006511497, -0.5101086582009359, 1.423174561540593, 0.8023474477536127, 1.6002930165500948]\n",
      "Iteration 15: [1.1051356392682812, 2.084900890623056, -1.4832217516081925, 0.5626626935604669, -0.5101095084741849, 1.4231808226736724, 0.8023467578090688, 1.6002934309692016]\n",
      "Iteration 16: [1.1051344393839442, 2.0849023460889717, -1.4832250162374057, 0.5626638390040378, -0.5101084067880209, 1.4231792995685275, 0.8023474253642874, 1.6002933447261336]\n",
      "Iteration 17: [1.1051352378563783, 2.0849016278304022, -1.4832240592380461, 0.5626632948923049, -0.5101086179843229, 1.4231805660500463, 0.802347264429466, 1.6002934281705359]\n",
      "Iteration 18: [1.1051349746306494, 2.084901935812856, -1.483224725114019, 0.562663536607202, -0.5101083970008448, 1.4231802206174957, 0.8023473994220582, 1.6002934080536833]\n",
      "Iteration 19: [1.1051351386040893, 2.0849017854268204, -1.4832245138693727, 0.5626634225372139, -0.510108446906423, 1.4231804780762425, 0.802347362867118, 1.6002934249277572]\n",
      "Iteration 20: [1.1051350813165566, 2.0849018506475523, -1.4832246503472872, 0.5626634735786453, -0.5101084023678113, 1.423180401066239, 0.8023473903003999, 1.6002934203583898]\n",
      "Iteration 21: [1.1051351151342126, 2.084901819112121, -1.4832246041605632, 0.5626634496118538, -0.5101084138098751, 1.4231804536606638, 0.8023473821424629, 1.60029342378755]\n",
      "Iteration 22: [1.1051351027433247, 2.0849018329216444, -1.4832246322526441, 0.5626634603910997, -0.510108404785202, 1.4231804367004401, 0.8023473877448215, 1.6002934227678078]\n",
      "Iteration 23: [1.1051351097426934, 2.0849018263015804, -1.4832246222275227, 0.5626634553497684, -0.5101084073564467, 1.423180447493235, 0.8023473859468246, 1.6002934234681028]\n",
      "Iteration 24: [1.1051351070756625, 2.0849018292241697, -1.4832246280313743, 0.5626634576259408, -0.5101084055181957, 1.42318044379165, 0.8023473870961337, 1.600293423243353]\n",
      "Iteration 25: [1.1051351085286918, 2.084901827833315, -1.4832246258676165, 0.5626634565648269, -0.5101084060878621, 1.4231804460153865, 0.8023473867035003, 1.6002934233870167]\n",
      "Iteration 26: [1.1051351079568548, 2.0849018284515086, -1.48322462707054, 0.5626634570453813, -0.5101084057116, 1.4231804452130625, 0.8023473869402403, 1.6002934233379376]\n",
      "Iteration 27: [1.1051351082592589, 2.084901828159106, -1.4832246266056006, 0.5626634568219445, -0.5101084058364993, 1.423180445672866, 0.8023473868551001, 1.60029342336753]\n",
      "Converged in 27 iterations\n",
      "Iteration 0: [0.6, 2.272727272727273, -1.1, 1.25, -0.8888888888888888, 1.0, 0.5, 1.5]\n",
      "Iteration 1: [1.0472727272727274, 1.8863636363636365, -1.1677272727272725, 0.516077441077441, -0.5277777777777778, 1.2022222222222223, 0.75, 1.5625]\n",
      "Iteration 2: [1.0221818181818183, 2.1210284664830117, -1.4298771043771044, 0.6371170033670034, -0.5643864571642349, 1.319762626262626, 0.7764722222222222, 1.59375]\n",
      "Iteration 3: [1.098078267523722, 2.061904882154882, -1.42455060453015, 0.5535542532508021, -0.5248175271230827, 1.3937330621025066, 0.7913512626262627, 1.5970590277777779]\n",
      "Iteration 4: [1.091100609121518, 2.092078627567379, -1.4761896585949281, 0.5720764351568434, -0.5176644025049094, 1.4015368021970547, 0.7990792089880284, 1.5989189078282828]\n",
      "Iteration 5: [1.1044457944757236, 2.0816983313687327, -1.472265656210998, 0.5608258380165022, -0.513872217827672, 1.4192319379762997, 0.8000455710025338, 1.5998849011235035]\n",
      "Iteration 6: [1.1026229643790728, 2.0863366022013836, -1.4824063233495113, 0.5640639276545943, -0.5111900317812109, 1.4189098103980184, 0.8019116839099804, 1.6000056963753166]\n",
      "Iteration 7: [1.1051149248900407, 2.084365896187798, -1.4811574830096224, 0.5622828198554272, -0.5109018279499299, 1.4226750590396091, 0.8018915506773336, 1.6002389604887475]\n",
      "Iteration 8: [1.1046680862207043, 2.0851917256649215, -1.4831606310855683, 0.5629035833730877, -0.5102630068961506, 1.422356034380634, 0.8022914019528358, 1.6002364438346668]\n",
      "Iteration 9: [1.1051512987836059, 2.0847997004559886, -1.4828308966545303, 0.562583432085293, -0.5102649275406272, 1.4231247281417239, 0.8022592478215301, 1.6002864252441045]\n",
      "Iteration 10: [1.105046149376505, 2.084960918715745, -1.48322936494511, 0.5627087562030731, -0.51012967907014, 1.4230222082703867, 0.8023411153385828, 1.6002824059776912]\n",
      "Iteration 11: [1.1051419648605965, 2.084880955983834, -1.483148924864535, 0.5626465166531261, -0.5101385363617948, 1.4231769852033633, 0.8023304614248078, 1.600292639417323]\n",
      "Iteration 12: [1.1051178805712905, 2.0849139536396986, -1.4832287412694325, 0.562672472568514, -0.5101110569933497, 1.4231500163294823, 0.8023469624620686, 1.600291307678101]\n",
      "Iteration 13: [1.1051371436178563, 2.0848974292360287, -1.4832099383922814, 0.5626598617348436, -0.5101141660858357, 1.4231811072283667, 0.8023441324007583, 1.6002933703077586]\n",
      "Iteration 14: [1.1051317306020592, 2.084904329092822, -1.4832260317949941, 0.5626653006511497, -0.5101086582009359, 1.423174561540593, 0.8023474477536127, 1.6002930165500948]\n",
      "Iteration 15: [1.1051356392682812, 2.084900890623056, -1.4832217516081925, 0.5626626935604669, -0.5101095084741849, 1.4231808226736724, 0.8023467578090688, 1.6002934309692016]\n",
      "Iteration 16: [1.1051344393839442, 2.0849023460889717, -1.4832250162374057, 0.5626638390040378, -0.5101084067880209, 1.4231792995685275, 0.8023474253642874, 1.6002933447261336]\n",
      "Iteration 17: [1.1051352378563783, 2.0849016278304022, -1.4832240592380461, 0.5626632948923049, -0.5101086179843229, 1.4231805660500463, 0.802347264429466, 1.6002934281705359]\n",
      "Iteration 18: [1.1051349746306494, 2.084901935812856, -1.483224725114019, 0.562663536607202, -0.5101083970008448, 1.4231802206174957, 0.8023473994220582, 1.6002934080536833]\n",
      "Iteration 19: [1.1051351386040893, 2.0849017854268204, -1.4832245138693727, 0.5626634225372139, -0.510108446906423, 1.4231804780762425, 0.802347362867118, 1.6002934249277572]\n",
      "Iteration 20: [1.1051350813165566, 2.0849018506475523, -1.4832246503472872, 0.5626634735786453, -0.5101084023678113, 1.423180401066239, 0.8023473903003999, 1.6002934203583898]\n",
      "Iteration 21: [1.1051351151342126, 2.084901819112121, -1.4832246041605632, 0.5626634496118538, -0.5101084138098751, 1.4231804536606638, 0.8023473821424629, 1.60029342378755]\n",
      "Iteration 22: [1.1051351027433247, 2.0849018329216444, -1.4832246322526441, 0.5626634603910997, -0.510108404785202, 1.4231804367004401, 0.8023473877448215, 1.6002934227678078]\n",
      "Iteration 23: [1.1051351097426934, 2.0849018263015804, -1.4832246222275227, 0.5626634553497684, -0.5101084073564467, 1.423180447493235, 0.8023473859468246, 1.6002934234681028]\n",
      "Iteration 24: [1.1051351070756625, 2.0849018292241697, -1.4832246280313743, 0.5626634576259408, -0.5101084055181957, 1.42318044379165, 0.8023473870961337, 1.600293423243353]\n",
      "Iteration 25: [1.1051351085286918, 2.084901827833315, -1.4832246258676165, 0.5626634565648269, -0.5101084060878621, 1.4231804460153865, 0.8023473867035003, 1.6002934233870167]\n",
      "Iteration 26: [1.1051351079568548, 2.0849018284515086, -1.48322462707054, 0.5626634570453813, -0.5101084057116, 1.4231804452130625, 0.8023473869402403, 1.6002934233379376]\n",
      "Iteration 27: [1.1051351082592589, 2.084901828159106, -1.4832246266056006, 0.5626634568219445, -0.5101084058364993, 1.423180445672866, 0.8023473868551001, 1.60029342336753]\n",
      "Converged in 27 iterations\n",
      "Final Solution: [1.1051351082592589, 2.084901828159106, -1.4832246266056006, 0.5626634568219445, -0.5101084058364993, 1.423180445672866, 0.8023473868551001, 1.60029342336753]\n",
      "Number of iterations: 27\n",
      "Time taken (seconds): 0.02499985694885254\n"
     ]
    }
   ],
   "source": [
    "# Hard-coded Example Input (8x8 Matrix)\n",
    "A_8x8 = [10.0  -1.0   2.0   0.0   0.0   0.0   0.0   0.0;\n",
    "         -1.0  11.0  -1.0   3.0   0.0   0.0   0.0   0.0;\n",
    "          2.0  -1.0  10.0  -1.0   0.0   3.0   0.0   0.0;\n",
    "          0.0   3.0  -1.0  12.0  -1.0   0.0   0.0   0.0;\n",
    "          0.0   0.0   0.0  -1.0   9.0  -2.0   0.0   0.0;\n",
    "          0.0   0.0   3.0   0.0  -2.0  10.0  -1.0   0.0;\n",
    "          0.0   0.0   0.0   0.0   0.0  -1.0  10.0  -1.0;\n",
    "          0.0   0.0   0.0   0.0   0.0   0.0  -1.0  8.0]\n",
    "\n",
    "b_8x8 = [6.0, 25.0, -11.0, 15.0, -8.0, 10.0, 5.0, 12.0]  # Vector b\n",
    "\n",
    "x0_8x8 = zeros(8)  # Initial guess, a vector of zeros (8 elements)\n",
    "\n",
    "# Call the Jacobi method for the 8x8 matrix\n",
    "x_sol_8x8, iterations_8x8 = jacobi(A_8x8, b_8x8, x0_8x8)\n",
    "\n",
    "println(\"8x8 Final Solution: \", x_sol_8x8)\n",
    "println(\"8x8 Number of iterations: \", iterations_8x8)\n",
    "\n",
    "# Start the timer\n",
    "start_time = time()\n",
    "\n",
    "# Call the Jacobi method for the 8x8 matrix\n",
    "x_sol_8x8, iterations_8x8 = jacobi(A_8x8, b_8x8, x0_8x8)\n",
    "function main()\n",
    "    # Start the timer\n",
    "    start_time = time()\n",
    "\n",
    "    # Call the Jacobi method with the matrix A_8x8, vector b_8x8, and initial guess x0_8x8\n",
    "    x_sol, iterations = jacobi(A_8x8, b_8x8, x0_8x8)\n",
    "\n",
    "    # End the timer\n",
    "    end_time = time()\n",
    "\n",
    "    # Calculate the duration\n",
    "    duration = end_time - start_time\n",
    "\n",
    "    # Output the final solution, number of iterations, and time taken\n",
    "    println(\"Final Solution: \", x_sol)\n",
    "    println(\"Number of iterations: \", iterations)\n",
    "    println(\"Time taken (seconds): \", duration)\n",
    "end\n",
    "\n",
    "# Call the main function\n",
    "main()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a6aa2709-ee91-4fb8-997a-51c91e5802a9",
   "metadata": {},
   "source": [
    "# Hard-coded Example Input (9x9 Matrix)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "e0ffaffc-6c4f-49c5-8045-2fca33d9dbfb",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0: [0.6, 2.272727272727273, -1.1, 1.25, -0.8888888888888888, 1.0, 0.5, 1.0, 1.0]\n",
      "Iteration 1: [1.0472727272727274, 1.8863636363636365, -1.1677272727272725, 0.516077441077441, -0.5277777777777778, 1.2022222222222223, 0.7, 1.125, 1.1428571428571428]\n",
      "Iteration 2: [1.0221818181818183, 2.1210284664830117, -1.4298771043771044, 0.6371170033670034, -0.5643864571642349, 1.3147626262626262, 0.7327222222222222, 1.1535714285714287, 1.1607142857142858]\n",
      "Iteration 3: [1.098078267523722, 2.061904882154882, -1.4230506045301499, 0.5535542532508021, -0.5259286382341938, 1.3893580621025066, 0.7468334054834055, 1.1577863756613758, 1.1647959183673469]\n",
      "Iteration 4: [1.090800609121518, 2.0922149912037424, -1.4748771585949279, 0.5721088425642509, -0.5186366247271317, 1.3964127942605467, 0.7547144437763882, 1.1593024436542294, 1.1653980536659108]\n",
      "Iteration 5: [1.1041969308393598, 2.0817815384394396, -1.4706515767256683, 0.5608201035888928, -0.5150072854349618, 1.4142072670106909, 0.7555715237914776, 1.1600093747868583, 1.1656146348077472]\n",
      "Iteration 6: [1.1023084691890777, 2.0864622766679104, -1.4808414020682459, 0.564083043543421, -0.5123072624877474, 1.4137511683098558, 0.7574216641797549, 1.1600988465499353, 1.165715624969551]\n",
      "Iteration 7: [1.1048145080804403, 2.084474357862779, -1.479532512309639, 0.5622887087866896, -0.5120460688707631, 1.4175331345408997, 0.7573850014859791, 1.160261440762442, 1.165728406649991]\n",
      "Iteration 8: [1.1043539382482057, 2.0853105335827937, -1.4815465353134112, 0.5629165287692718, -0.5114050024590568, 1.417189040067337, 0.7577794575303342, 1.1602594506779975, 1.1657516343946346]\n",
      "Iteration 9: [1.1048403604209616, 2.084914346966089, -1.4812047934346357, 0.5625930717899292, -0.5114117101217838, 1.4179609058552454, 0.7577448490745334, 1.160294257660414, 1.1657513500968568]\n",
      "Iteration 10: [1.1047323933835362, 2.085077850146958, -1.481605601965164, 0.5627200379621095, -0.5112761240555089, 1.4178535809134873, 0.7578255163515659, 1.1602913499309493, 1.1657563225229164]\n",
      "Iteration 11: [1.1048289054077285, 2.0849969706847316, -1.4815227641398467, 0.5626570602948711, -0.5112858666901017, 1.418009007413604, 0.7578144930844436, 1.1602984865728736, 1.1657559071329928]\n",
      "Iteration 12: [1.1048042498964423, 2.0850304509439335, -1.4816030802076667, 0.5626833714263214, -0.5112583249864356, 1.417981105212378, 0.7578307493986477, 1.160297533351453, 1.1657569266532677]\n",
      "Iteration 13: [1.1048236611359266, 2.085013732309983, -1.4815837993059762, 0.5626706034978414, -0.5112616020165469, 1.4180123340048776, 0.7578278638563831, 1.1602989730043263, 1.1657567904787791]\n",
      "Iteration 14: [1.1048181330921936, 2.085020731939675, -1.4815999988478663, 0.5626761168122941, -0.5112560809436003, 1.4180056057741217, 0.7578311307009205, 1.1602987211945968, 1.1657569961434753]\n",
      "Iteration 15: [1.1048220729635407, 2.085017253073404, -1.4815956234754784, 0.5626734770324591, -0.5112569635154958, 1.418011896535732, 0.7578304326968718, 1.1602990105703663, 1.1657569601706566]\n",
      "Iteration 16: [1.1048208500024361, 2.085018728944608, -1.4815989105428415, 0.5626746378157345, -0.5112558588773418, 1.4180103376092315, 0.7578310907106098, 1.1602989494056273, 1.1657570015100522]\n",
      "Iteration 17: [1.1048216550030292, 2.085018002364763, -1.4815979346072223, 0.5626740869788328, -0.5112560763295336, 1.418011610458445, 0.7578309287014859, 1.1602990076850552, 1.1657569927722324]\n",
      "Iteration 18: [1.1048213871579207, 2.085018314496301, -1.48159860520378, 0.5626743318307462, -0.511255854678253, 1.4180112579864086, 0.75783106181435, 1.1602989934561432, 1.1657570010978648]\n",
      "Iteration 19: [1.104821552490386, 2.0850181624056274, -1.481598390194802, 0.5626742163857553, -0.511255905799604, 1.4180115168069185, 0.7578310251442553, 1.1602990052426845, 1.1657569990651633]\n",
      "Iteration 20: [1.1048214942795231, 2.08501822846712, -1.4815985276610144, 0.562674268065726, -0.5112558611111564, 1.4180114384129454, 0.7578310522049603, 1.1602990020174515, 1.165757000748955]\n",
      "Iteration 21: [1.104821528378915, 2.0850181965837575, -1.4815984807265037, 0.5626742438188724, -0.5112558727898203, 1.4180114912965691, 0.7578310440430397, 1.1602990044128263, 1.1657570002882074]\n",
      "Iteration 22: [1.1048215158036765, 2.0850182105632538, -1.4815985090244905, 0.5626742547277002, -0.5112558637319987, 1.418011474064291, 0.7578310495709395, 1.1602990036942706, 1.1657570006304037]\n",
      "Iteration 23: [1.1048215228612235, 2.0850182038723712, -1.4815984988509272, 0.5626742496294792, -0.511255866349302, 1.4180114849180414, 0.7578310477758562, 1.1602990041834451, 1.1657570005277529]\n",
      "Iteration 24: [1.1048215201574227, 2.0850182068292598, -1.481598504697472, 0.5626742519318881, -0.5112558645038264, 1.4180114811630034, 0.7578310489101486, 1.1602990040253007, 1.165757000597635]\n",
      "Iteration 25: [1.1048215216224204, 2.085018205424026, -1.4815985025042706, 0.5626742508592435, -0.5112558650824561, 1.418011483399491, 0.7578310485188304, 1.1602990041256487, 1.165757000575043]\n",
      "Iteration 26: [1.1048215210432566, 2.085018206049129, -1.4815985037160044, 0.5626742513450996, -0.5112558647046416, 1.418011482586673, 0.757831048752514, 1.160299004091156, 1.1657570005893785]\n",
      "Iteration 27: [1.1048215213481138, 2.085018205753814, -1.4815985032452303, 0.5626742511193306, -0.5112558648312837, 1.4180114830491246, 0.7578310486677828, 1.1602990041118244, 1.1657570005844509]\n",
      "Converged in 27 iterations\n",
      "Iteration 0: [0.6, 2.272727272727273, -1.1, 1.25, -0.8888888888888888, 1.0, 0.5, 1.0, 1.0]\n",
      "Iteration 1: [1.0472727272727274, 1.8863636363636365, -1.1677272727272725, 0.516077441077441, -0.5277777777777778, 1.2022222222222223, 0.7, 1.125, 1.1428571428571428]\n",
      "Iteration 2: [1.0221818181818183, 2.1210284664830117, -1.4298771043771044, 0.6371170033670034, -0.5643864571642349, 1.3147626262626262, 0.7327222222222222, 1.1535714285714287, 1.1607142857142858]\n",
      "Iteration 3: [1.098078267523722, 2.061904882154882, -1.4230506045301499, 0.5535542532508021, -0.5259286382341938, 1.3893580621025066, 0.7468334054834055, 1.1577863756613758, 1.1647959183673469]\n",
      "Iteration 4: [1.090800609121518, 2.0922149912037424, -1.4748771585949279, 0.5721088425642509, -0.5186366247271317, 1.3964127942605467, 0.7547144437763882, 1.1593024436542294, 1.1653980536659108]\n",
      "Iteration 5: [1.1041969308393598, 2.0817815384394396, -1.4706515767256683, 0.5608201035888928, -0.5150072854349618, 1.4142072670106909, 0.7555715237914776, 1.1600093747868583, 1.1656146348077472]\n",
      "Iteration 6: [1.1023084691890777, 2.0864622766679104, -1.4808414020682459, 0.564083043543421, -0.5123072624877474, 1.4137511683098558, 0.7574216641797549, 1.1600988465499353, 1.165715624969551]\n",
      "Iteration 7: [1.1048145080804403, 2.084474357862779, -1.479532512309639, 0.5622887087866896, -0.5120460688707631, 1.4175331345408997, 0.7573850014859791, 1.160261440762442, 1.165728406649991]\n",
      "Iteration 8: [1.1043539382482057, 2.0853105335827937, -1.4815465353134112, 0.5629165287692718, -0.5114050024590568, 1.417189040067337, 0.7577794575303342, 1.1602594506779975, 1.1657516343946346]\n",
      "Iteration 9: [1.1048403604209616, 2.084914346966089, -1.4812047934346357, 0.5625930717899292, -0.5114117101217838, 1.4179609058552454, 0.7577448490745334, 1.160294257660414, 1.1657513500968568]\n",
      "Iteration 10: [1.1047323933835362, 2.085077850146958, -1.481605601965164, 0.5627200379621095, -0.5112761240555089, 1.4178535809134873, 0.7578255163515659, 1.1602913499309493, 1.1657563225229164]\n",
      "Iteration 11: [1.1048289054077285, 2.0849969706847316, -1.4815227641398467, 0.5626570602948711, -0.5112858666901017, 1.418009007413604, 0.7578144930844436, 1.1602984865728736, 1.1657559071329928]\n",
      "Iteration 12: [1.1048042498964423, 2.0850304509439335, -1.4816030802076667, 0.5626833714263214, -0.5112583249864356, 1.417981105212378, 0.7578307493986477, 1.160297533351453, 1.1657569266532677]\n",
      "Iteration 13: [1.1048236611359266, 2.085013732309983, -1.4815837993059762, 0.5626706034978414, -0.5112616020165469, 1.4180123340048776, 0.7578278638563831, 1.1602989730043263, 1.1657567904787791]\n",
      "Iteration 14: [1.1048181330921936, 2.085020731939675, -1.4815999988478663, 0.5626761168122941, -0.5112560809436003, 1.4180056057741217, 0.7578311307009205, 1.1602987211945968, 1.1657569961434753]\n",
      "Iteration 15: [1.1048220729635407, 2.085017253073404, -1.4815956234754784, 0.5626734770324591, -0.5112569635154958, 1.418011896535732, 0.7578304326968718, 1.1602990105703663, 1.1657569601706566]\n",
      "Iteration 16: [1.1048208500024361, 2.085018728944608, -1.4815989105428415, 0.5626746378157345, -0.5112558588773418, 1.4180103376092315, 0.7578310907106098, 1.1602989494056273, 1.1657570015100522]\n",
      "Iteration 17: [1.1048216550030292, 2.085018002364763, -1.4815979346072223, 0.5626740869788328, -0.5112560763295336, 1.418011610458445, 0.7578309287014859, 1.1602990076850552, 1.1657569927722324]\n",
      "Iteration 18: [1.1048213871579207, 2.085018314496301, -1.48159860520378, 0.5626743318307462, -0.511255854678253, 1.4180112579864086, 0.75783106181435, 1.1602989934561432, 1.1657570010978648]\n",
      "Iteration 19: [1.104821552490386, 2.0850181624056274, -1.481598390194802, 0.5626742163857553, -0.511255905799604, 1.4180115168069185, 0.7578310251442553, 1.1602990052426845, 1.1657569990651633]\n",
      "Iteration 20: [1.1048214942795231, 2.08501822846712, -1.4815985276610144, 0.562674268065726, -0.5112558611111564, 1.4180114384129454, 0.7578310522049603, 1.1602990020174515, 1.165757000748955]\n",
      "Iteration 21: [1.104821528378915, 2.0850181965837575, -1.4815984807265037, 0.5626742438188724, -0.5112558727898203, 1.4180114912965691, 0.7578310440430397, 1.1602990044128263, 1.1657570002882074]\n",
      "Iteration 22: [1.1048215158036765, 2.0850182105632538, -1.4815985090244905, 0.5626742547277002, -0.5112558637319987, 1.418011474064291, 0.7578310495709395, 1.1602990036942706, 1.1657570006304037]\n",
      "Iteration 23: [1.1048215228612235, 2.0850182038723712, -1.4815984988509272, 0.5626742496294792, -0.511255866349302, 1.4180114849180414, 0.7578310477758562, 1.1602990041834451, 1.1657570005277529]\n",
      "Iteration 24: [1.1048215201574227, 2.0850182068292598, -1.481598504697472, 0.5626742519318881, -0.5112558645038264, 1.4180114811630034, 0.7578310489101486, 1.1602990040253007, 1.165757000597635]\n",
      "Iteration 25: [1.1048215216224204, 2.085018205424026, -1.4815985025042706, 0.5626742508592435, -0.5112558650824561, 1.418011483399491, 0.7578310485188304, 1.1602990041256487, 1.165757000575043]\n",
      "Iteration 26: [1.1048215210432566, 2.085018206049129, -1.4815985037160044, 0.5626742513450996, -0.5112558647046416, 1.418011482586673, 0.757831048752514, 1.160299004091156, 1.1657570005893785]\n",
      "Iteration 27: [1.1048215213481138, 2.085018205753814, -1.4815985032452303, 0.5626742511193306, -0.5112558648312837, 1.4180114830491246, 0.7578310486677828, 1.1602990041118244, 1.1657570005844509]\n",
      "Converged in 27 iterations\n",
      "Final Solution: [1.1048215213481138, 2.085018205753814, -1.4815985032452303, 0.5626742511193306, -0.5112558648312837, 1.4180114830491246, 0.7578310486677828, 1.1602990041118244, 1.1657570005844509]\n",
      "Number of iterations: 27\n",
      "Time taken (seconds): 0.029999971389770508\n"
     ]
    }
   ],
   "source": [
    "# Hard-coded Example Input (9x9 Matrix)\n",
    "A_9x9 = [10.0  -1.0   2.0   0.0   0.0   0.0   0.0   0.0   0.0;\n",
    "         -1.0  11.0  -1.0   3.0   0.0   0.0   0.0   0.0   0.0;\n",
    "          2.0  -1.0  10.0  -1.0   0.0   3.0   0.0   0.0   0.0;\n",
    "          0.0   3.0  -1.0  12.0  -1.0   0.0   0.0   0.0   0.0;\n",
    "          0.0   0.0   0.0  -1.0   9.0  -2.0   0.0   0.0   0.0;\n",
    "          0.0   0.0   3.0   0.0  -2.0  10.0  -1.0   0.0   0.0;\n",
    "          0.0   0.0   0.0   0.0   0.0  -1.0  10.0  -1.0   0.0;\n",
    "          0.0   0.0   0.0   0.0   0.0   0.0  -1.0  12.0  -1.0;\n",
    "          0.0   0.0   0.0   0.0   0.0   0.0   0.0  -1.0   7.0]\n",
    "\n",
    "b_9x9 = [6.0, 25.0, -11.0, 15.0, -8.0, 10.0, 5.0, 12.0, 7.0]  # Vector b\n",
    "\n",
    "x0_9x9 = zeros(9)  # Initial guess, a vector of zeros (9 elements)\n",
    "\n",
    "# Call the Jacobi method for the 9x9 matrix\n",
    "x_sol_9x9, iterations_9x9 = jacobi(A_9x9, b_9x9, x0_9x9)\n",
    "\n",
    "function main()\n",
    "    # Start the timer\n",
    "    start_time = time()\n",
    "\n",
    "    # Call the Jacobi method with the matrix A_9x9, vector b_9x9, and initial guess x0_9x9\n",
    "    x_sol, iterations = jacobi(A_9x9, b_9x9, x0_9x9)\n",
    "\n",
    "    # End the timer\n",
    "    end_time = time()\n",
    "\n",
    "    # Calculate the duration\n",
    "    duration = end_time - start_time\n",
    "\n",
    "    # Output the final solution, number of iterations, and time taken\n",
    "    println(\"Final Solution: \", x_sol)\n",
    "    println(\"Number of iterations: \", iterations)\n",
    "    println(\"Time taken (seconds): \", duration)\n",
    "end\n",
    "\n",
    "# Call the main function\n",
    "main()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "09754283-72c3-40a6-96ae-958f574683f1",
   "metadata": {},
   "source": [
    "# Function to perform Gauss-Seidel Iteration for fixed point iteration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "26fd2592-55c9-4399-bc08-1eb5e7fefb0e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "read_input (generic function with 1 method)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using LinearAlgebra\n",
    "\n",
    "# Function to perform Gauss-Seidel Iteration and print all intermediate results\n",
    "function gauss_seidel(A, b, x0; tol=1e-9, max_iter=1000)\n",
    "    n = length(b)               # Number of unknowns\n",
    "    x = copy(x0)                # Initial guess\n",
    "    iter = 0                    # Iteration counter\n",
    "\n",
    "    while iter < max_iter\n",
    "        for i in 1:n\n",
    "            sum = 0.0\n",
    "            for j in 1:n\n",
    "                if i != j\n",
    "                    sum += A[i, j] * x[j]  # Sum of off-diagonal elements\n",
    "                end\n",
    "            end\n",
    "            # Update x[i] using the new value immediately (Gauss-Seidel specific)\n",
    "            x[i] = (b[i] - sum) / A[i, i]\n",
    "        end\n",
    "\n",
    "        # Print current iteration and solution\n",
    "        println(\"Iteration $iter: \", x)\n",
    "\n",
    "        # Check for convergence (if the difference between A*x and b is below tolerance)\n",
    "        if norm(A * x - b, Inf) < tol\n",
    "            println(\"Converged in $iter iterations\")\n",
    "            return x, iter\n",
    "        end\n",
    "\n",
    "        iter += 1\n",
    "    end\n",
    "\n",
    "    println(\"Maximum iterations reached. Did not converge.\")\n",
    "    return x, iter\n",
    "end\n",
    "\n",
    "# Input function to read matrix A, vector b, and initial guess x0\n",
    "function read_input()\n",
    "    # Reading matrix A dimensions\n",
    "    println(\"Enter the size of the matrix A (number of rows or columns, for a square matrix):\")\n",
    "    \n",
    "    # Input validation for matrix size\n",
    "    n = 0\n",
    "    while n <= 0\n",
    "        input = readline()\n",
    "        if isempty(input)\n",
    "            println(\"Input cannot be empty. Please enter a valid number:\")\n",
    "        else\n",
    "            try\n",
    "                n = parse(Int, input)\n",
    "                if n <= 0\n",
    "                    println(\"Size of matrix must be a positive integer. Please enter a valid number:\")\n",
    "                end\n",
    "            catch\n",
    "                println(\"Invalid input. Please enter a valid integer:\")\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "\n",
    "    # Initialize an empty matrix A and vector b\n",
    "    A = zeros(n, n)\n",
    "    b = zeros(n)\n",
    "\n",
    "    # Read matrix A with input validation\n",
    "    println(\"Enter the elements of matrix A row by row (space-separated values for each row):\")\n",
    "    for i in 1:n\n",
    "        row_valid = false\n",
    "        while !row_valid\n",
    "            input = readline()\n",
    "            row_values = split(input)\n",
    "            if length(row_values) == n\n",
    "                try\n",
    "                    A[i, :] = parse.(Float64, row_values)\n",
    "                    row_valid = true\n",
    "                catch\n",
    "                    println(\"Invalid input. Please enter valid floating point numbers for row $i:\")\n",
    "                end\n",
    "            else\n",
    "                println(\"Incorrect number of elements. Please enter $n elements for row $i:\")\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "\n",
    "    # Read vector b with input validation\n",
    "    println(\"Enter the elements of vector b (space-separated values):\")\n",
    "    vec_valid = false\n",
    "    while !vec_valid\n",
    "        input = readline()\n",
    "        vec_values = split(input)\n",
    "        if length(vec_values) == n\n",
    "            try\n",
    "                b = parse.(Float64, vec_values)\n",
    "                vec_valid = true\n",
    "            catch\n",
    "                println(\"Invalid input. Please enter valid floating point numbers for vector b:\")\n",
    "            end\n",
    "        else\n",
    "            println(\"Incorrect number of elements. Please enter $n elements for vector b:\")\n",
    "        end\n",
    "    end\n",
    "\n",
    "    # Read initial guess x0 with input validation\n",
    "    println(\"Enter the initial guess for vector x0 (space-separated values):\")\n",
    "    x0_valid = false\n",
    "    while !x0_valid\n",
    "        input = readline()\n",
    "        x0_values = split(input)\n",
    "        if length(x0_values) == n\n",
    "            try\n",
    "                x0 = parse.(Float64, x0_values)\n",
    "                x0_valid = true\n",
    "            catch\n",
    "                println(\"Invalid input. Please enter valid floating point numbers for vector x0:\")\n",
    "            end\n",
    "        else\n",
    "            println(\"Incorrect number of elements. Please enter $n elements for vector x0:\")\n",
    "        end\n",
    "    end\n",
    "\n",
    "    return A, b, x0\n",
    "end\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b465d958-9fbb-4560-b783-30aac97a2940",
   "metadata": {},
   "source": [
    "# Hard-coded Example Input (2x2 Matrix)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "845c9f5b-5020-4ed8-b58c-210a6cda73dd",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0: [3.0, 2.5]\n",
      "Iteration 1: [2.1666666666666665, 2.916666666666667]\n",
      "Iteration 2: [2.0277777777777777, 2.986111111111111]\n",
      "Iteration 3: [2.0046296296296298, 2.997685185185185]\n",
      "Iteration 4: [2.0007716049382718, 2.999614197530864]\n",
      "Iteration 5: [2.000128600823045, 2.9999356995884776]\n",
      "Iteration 6: [2.0000214334705073, 2.999989283264746]\n",
      "Iteration 7: [2.0000035722450846, 2.9999982138774577]\n",
      "Iteration 8: [2.0000005953741806, 2.99999970231291]\n",
      "Iteration 9: [2.00000009922903, 2.999999950385485]\n",
      "Iteration 10: [2.0000000165381717, 2.999999991730914]\n",
      "Iteration 11: [2.000000002756362, 2.999999998621819]\n",
      "Iteration 12: [2.000000000459394, 2.999999999770303]\n",
      "Iteration 13: [2.000000000076566, 2.999999999961717]\n",
      "Converged in 13 iterations\n",
      "Final Solution: [2.000000000076566, 2.999999999961717]\n",
      "Number of iterations: 13\n",
      "Time taken (seconds): 0.005000114440917969\n"
     ]
    }
   ],
   "source": [
    "# Hard-coded Example Input (2x2 Matrix)\n",
    "A_2x2 = [3.0  1.0;\n",
    "         1.0  2.0]\n",
    "\n",
    "b_2x2 = [9.0, 8.0]\n",
    "\n",
    "x0_2x2 = zeros(2)\n",
    "\n",
    "function main()\n",
    "    start_time = time()\n",
    "    x_sol, iterations = gauss_seidel(A_2x2, b_2x2, x0_2x2)\n",
    "    end_time = time()\n",
    "    duration = end_time - start_time\n",
    "    println(\"Final Solution: \", x_sol)\n",
    "    println(\"Number of iterations: \", iterations)\n",
    "    println(\"Time taken (seconds): \", duration)\n",
    "end\n",
    "\n",
    "main()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d61ed649-2d7d-48a9-be69-69843bb1fcfe",
   "metadata": {},
   "source": [
    "# Hard-coded Example Input (3x3 Matrix)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "9cd66e9c-37ef-4b95-a93d-838f28349ac6",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0: [1.75, 1.9166666666666667, 0.08333333333333322]\n",
      "Iteration 1: [2.2083333333333335, 2.013888888888889, -0.047222222222222145]\n",
      "Iteration 2: [2.2652777777777775, 2.1199074074074074, -0.10101851851851844]\n",
      "Iteration 3: [2.3052314814814814, 2.1690895061728397, -0.12868209876543216]\n",
      "Iteration 4: [2.324442901234568, 2.1939356995884776, -0.14246286008230466]\n",
      "Iteration 5: [2.3340996399176954, 2.2063417866941015, -0.1493566426611796]\n",
      "Iteration 6: [2.33892460733882, 2.2125459642203933, -0.15280330715592144]\n",
      "Iteration 7: [2.341337317844079, 2.215647977385307, -0.1545266545229385]\n",
      "Iteration 8: [2.3425436579770613, 2.217198989007646, -0.15538832719847057]\n",
      "Iteration 9: [2.343146829051529, 2.2179744944828235, -0.15581916360343512]\n",
      "Iteration 10: [2.3434484145215646, 2.2183622472428115, -0.15603458180143762]\n",
      "Iteration 11: [2.3435992072610623, 2.2185561236213123, -0.15614229090073728]\n",
      "Iteration 12: [2.3436746036305123, 2.2186530618106626, -0.1561961454503674]\n",
      "Iteration 13: [2.3437123018152572, 2.2187015309053306, -0.1562230727251837]\n",
      "Iteration 14: [2.3437311509076286, 2.2187257654526653, -0.15623653636259185]\n",
      "Iteration 15: [2.3437405754538143, 2.2187378827263324, -0.15624326818129575]\n",
      "Iteration 16: [2.343745287726907, 2.2187439413631664, -0.15624663409064787]\n",
      "Iteration 17: [2.3437476438634537, 2.218746970681583, -0.15624831704532394]\n",
      "Iteration 18: [2.343748821931727, 2.2187484853407917, -0.15624915852266205]\n",
      "Iteration 19: [2.3437494109658634, 2.218749242670396, -0.1562495792613312]\n",
      "Iteration 20: [2.343749705482932, 2.218749621335198, -0.1562497896306656]\n",
      "Iteration 21: [2.3437498527414657, 2.218749810667599, -0.1562498948153328]\n",
      "Iteration 22: [2.343749926370733, 2.2187499053337993, -0.15624994740766632]\n",
      "Iteration 23: [2.3437499631853664, 2.2187499526668994, -0.15624997370383298]\n",
      "Iteration 24: [2.343749981592683, 2.21874997633345, -0.1562499868519165]\n",
      "Iteration 25: [2.3437499907963417, 2.2187499881667248, -0.15624999342595824]\n",
      "Iteration 26: [2.343749995398171, 2.2187499940833626, -0.1562499967129792]\n",
      "Iteration 27: [2.3437499976990854, 2.2187499970416815, -0.15624999835648978]\n",
      "Iteration 28: [2.343749998849543, 2.2187499985208405, -0.1562499991782449]\n",
      "Iteration 29: [2.3437499994247712, 2.2187499992604205, -0.15624999958912245]\n",
      "Iteration 30: [2.3437499997123856, 2.21874999963021, -0.15624999979456114]\n",
      "Converged in 30 iterations\n",
      "Final Solution: [2.3437499997123856, 2.21874999963021, -0.15624999979456114]\n",
      "Number of iterations: 30\n",
      "Time taken (seconds): 0.01100015640258789\n"
     ]
    }
   ],
   "source": [
    "# Hard-coded Example Input (3x3 Matrix)\n",
    "A_3x3 = [4.0  -1.0   1.0;\n",
    "         -1.0  3.0  2.0;\n",
    "          1.0  2.0  5.0]\n",
    "\n",
    "b_3x3 = [7.0, 4.0, 6.0]\n",
    "\n",
    "x0_3x3 = zeros(3)\n",
    "\n",
    "function main()\n",
    "    start_time = time()\n",
    "    x_sol, iterations = gauss_seidel(A_3x3, b_3x3, x0_3x3)\n",
    "    end_time = time()\n",
    "    duration = end_time - start_time\n",
    "    println(\"Final Solution: \", x_sol)\n",
    "    println(\"Number of iterations: \", iterations)\n",
    "    println(\"Time taken (seconds): \", duration)\n",
    "end\n",
    "\n",
    "main()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "deeedbf3-b0a4-4532-a5de-50dd98e2b410",
   "metadata": {},
   "source": [
    "# Hard-coded Example Input (4x4 Matrix)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "c8ab130b-bf6d-46e9-b62f-dd5cf7815ff0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0: [0.6, 2.3272727272727276, -0.9872727272727273, 0.8788636363636363]\n",
      "Iteration 1: [1.0301818181818183, 2.0369380165289255, -1.0144561983471074, 0.9843412190082644]\n",
      "Iteration 2: [1.006585041322314, 2.003555016904583, -1.002527384673178, 0.9983509455766342]\n",
      "Iteration 3: [1.000860978625094, 2.000298250656547, -1.0003072761017007, 0.9998497464910823]\n",
      "Iteration 4: [1.000091280285995, 2.000021342246459, -1.0000311471834449, 0.9999881032596473]\n",
      "Iteration 5: [1.0000083636613348, 2.000001173336268, -1.0000027450726754, 0.9999992168648151]\n",
      "Iteration 6: [1.000000666348162, 2.0000000246073673, -1.000000209122414, 0.9999999646319355]\n",
      "Iteration 7: [1.0000000442852195, 1.9999999946606364, -1.0000000129277866, 1.000000000386288]\n",
      "Iteration 8: [1.0000000020516209, 1.9999999989059063, -1.0000000004811047, 1.000000000350147]\n",
      "Iteration 9: [0.9999999999868117, 1.9999999998595697, -0.9999999999763907, 1.0000000000556126]\n",
      "Iteration 10: [0.9999999999812351, 1.9999999999852733, -0.9999999999921585, 1.0000000000065028]\n",
      "Converged in 10 iterations\n",
      "Final Solution: [0.9999999999812351, 1.9999999999852733, -0.9999999999921585, 1.0000000000065028]\n",
      "Number of iterations: 10\n",
      "Time taken (seconds): 0.006000041961669922\n"
     ]
    }
   ],
   "source": [
    "# Hard-coded Example Input (4x4 Matrix)\n",
    "A_4x4 = [10.0   -1.0   2.0   0.0;\n",
    "         -1.0   11.0  -1.0   3.0;\n",
    "         2.0  -1.0   10.0  -1.0;\n",
    "         0.0   3.0  -1.0   8.0]\n",
    "\n",
    "b_4x4 = [6.0, 25.0, -11.0, 15.0]\n",
    "\n",
    "x0_4x4 = zeros(4)\n",
    "\n",
    "function main()\n",
    "    start_time = time()\n",
    "    x_sol, iterations = gauss_seidel(A_4x4, b_4x4, x0_4x4)\n",
    "    end_time = time()\n",
    "    duration = end_time - start_time\n",
    "    println(\"Final Solution: \", x_sol)\n",
    "    println(\"Number of iterations: \", iterations)\n",
    "    println(\"Time taken (seconds): \", duration)\n",
    "end\n",
    "\n",
    "main()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2261b9e5-6006-40bb-a2f1-7d267333c212",
   "metadata": {},
   "source": [
    "# Hard-coded Example Input (5x5 Matrix)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "f9e138f8-384e-4a98-8393-e9b551ad41e2",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0: [0.6, 2.3272727272727276, -0.9872727272727273, 0.5859090909090908, -0.8237878787878787]\n",
      "Iteration 1: [1.0301818181818183, 2.116834710743802, -1.0357619834710745, 0.5658288337924701, -0.8260190184675033]\n",
      "Iteration 2: [1.0188358677685951, 2.1168715802654643, -1.0354971321479256, 0.5656557590490148, -0.8260382489945539]\n",
      "Iteration 3: [1.0187865844561315, 2.1169383795601053, -1.0354979030303144, 0.5656373924412347, -0.8260402897287517]\n",
      "Iteration 4: [1.0187934185620733, 2.1169439398370957, -1.0355005504845818, 0.565635611689615, -0.8260404875900428]\n",
      "Iteration 5: [1.0187945040806259, 2.1169442835024728, -1.0355009112969165, 0.5656354792171353, -0.8260405023092071]\n",
      "Iteration 6: [1.0187946106096306, 2.116944296514664, -1.0355009445487462, 0.5656354719665045, -0.8260405031148328]\n",
      "Iteration 7: [1.0187946185612158, 2.116944296192087, -1.0355009468963838, 0.5656354717843769, -0.8260405031350692]\n",
      "Iteration 8: [1.0187946189984856, 2.116944296068088, -1.0355009470144505, 0.5656354718038513, -0.8260405031329054]\n",
      "Converged in 8 iterations\n",
      "Final Solution: [1.0187946189984856, 2.116944296068088, -1.0355009470144505, 0.5656354718038513, -0.8260405031329054]\n",
      "Number of iterations: 8\n",
      "Time taken (seconds): 0.009999990463256836\n"
     ]
    }
   ],
   "source": [
    "# Hard-coded Example Input (5x5 Matrix)\n",
    "A_5x5 = [10.0  -1.0   2.0   0.0   0.0;\n",
    "         -1.0  11.0  -1.0   3.0   0.0;\n",
    "          2.0  -1.0  10.0  -1.0   0.0;\n",
    "          0.0   3.0  -1.0  12.0  -1.0;\n",
    "          0.0   0.0   0.0  -1.0   9.0]\n",
    "\n",
    "b_5x5 = [6.0, 25.0, -11.0, 15.0, -8.0]\n",
    "\n",
    "x0_5x5 = zeros(5)\n",
    "\n",
    "function main()\n",
    "    start_time = time()\n",
    "    x_sol, iterations = gauss_seidel(A_5x5, b_5x5, x0_5x5)\n",
    "    end_time = time()\n",
    "    duration = end_time - start_time\n",
    "    println(\"Final Solution: \", x_sol)\n",
    "    println(\"Number of iterations: \", iterations)\n",
    "    println(\"Time taken (seconds): \", duration)\n",
    "end\n",
    "\n",
    "main()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6f671e26-6565-464b-aca2-381a10909285",
   "metadata": {},
   "source": [
    "# Hard-coded Example Input (6x6 Matrix)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "07c8edd8-b493-4d79-b70c-99f8780458a3",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0: [0.6, 2.3272727272727276, -0.9872727272727273, 0.5859090909090908, -0.8237878787878787, 1.1314242424242424]\n",
      "Iteration 1: [1.0301818181818183, 2.116834710743802, -1.3751892561983472, 0.537543227731864, -0.577734254157739, 1.2970099260279564]\n",
      "Iteration 2: [1.0867213223140495, 2.0999002166291008, -1.4427028978351004, 0.5566551831766549, -0.5388138849741592, 1.3250480923556984]\n",
      "Iteration 3: [1.09853060122993, 2.0896238321695333, -1.4525926464180767, 0.5616434976749303, -0.532028924179297, 1.3293720090895635]\n",
      "Iteration 4: [1.0994809125005687, 2.0874507066416093, -1.4537983647953288, 0.5626517159250456, -0.5309560295439808, 1.3299483035298025]\n",
      "Iteration 5: [1.0995047436232266, 2.0870682937320693, -1.4539134388178745, 0.5628271375368281, -0.5308084728226186, 1.3300123370808385]\n",
      "Iteration 6: [1.099489517136782, 2.087008605973493, -1.4539180302005756, 0.5628539732546938, -0.530791261398181, 1.3300171567805363]\n",
      "Iteration 7: [1.0994844666374646, 2.0870004106066187, -1.4539166019755225, 0.5628575754005367, -0.5307897901153767, 1.3300170225695815]\n",
      "Iteration 8: [1.0994833614557664, 2.086999457570785, -1.4539160757648957, 0.5628579801172812, -0.5307897749715063, 1.3300168677351674]\n",
      "Iteration 9: [1.0994831609100575, 2.0869993767993926, -1.4539159568108944, 0.5628580114849518, -0.530789805893857, 1.3300168258644969]\n",
      "Iteration 10: [1.099483129042118, 2.0869993761614882, -1.4539159348031288, 0.5628580109015457, -0.5307898152632734, 1.330016817388284]\n",
      "Iteration 11: [1.0994831245767744, 2.0869993779153644, -1.453915931250149, 0.5628580099783737, -0.530789817249451, 1.3300168159251544]\n",
      "Iteration 12: [1.0994831240415661, 2.0869993784414813, -1.453915930743874, 0.5628580097235192, -0.530789817602908, 1.3300168157025805]\n",
      "Iteration 13: [1.099483123992923, 2.08699937855259, -1.4539159306817477, 0.5628580096714645, -0.5307898176581527, 1.3300168156728938]\n",
      "Converged in 13 iterations\n",
      "Final Solution: [1.099483123992923, 2.08699937855259, -1.4539159306817477, 0.5628580096714645, -0.5307898176581527, 1.3300168156728938]\n",
      "Number of iterations: 13\n",
      "Time taken (seconds): 0.01100015640258789\n"
     ]
    }
   ],
   "source": [
    "# Hard-coded Example Input (6x6 Matrix)\n",
    "A_6x6 = [10.0  -1.0   2.0   0.0   0.0   0.0;\n",
    "         -1.0  11.0  -1.0   3.0   0.0   0.0;\n",
    "          2.0  -1.0  10.0  -1.0   0.0   3.0;\n",
    "          0.0   3.0  -1.0  12.0  -1.0   0.0;\n",
    "          0.0   0.0   0.0  -1.0   9.0  -2.0;\n",
    "          0.0   0.0   3.0   0.0  -2.0  10.0]\n",
    "\n",
    "b_6x6 = [6.0, 25.0, -11.0, 15.0, -8.0, 10.0]\n",
    "\n",
    "x0_6x6 = zeros(6)\n",
    "\n",
    "function main()\n",
    "    start_time = time()\n",
    "    x_sol, iterations = gauss_seidel(A_6x6, b_6x6, x0_6x6)\n",
    "    end_time = time()\n",
    "    duration = end_time - start_time\n",
    "    println(\"Final Solution: \", x_sol)\n",
    "    println(\"Number of iterations: \", iterations)\n",
    "    println(\"Time taken (seconds): \", duration)\n",
    "end\n",
    "\n",
    "main()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8a9b2ab6-499f-43e8-ab01-4b57741b3a89",
   "metadata": {},
   "source": [
    "# Hard-coded Example Input (7x7 Matrix)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "f39356bb-9795-4245-a77c-ba92ac083e6e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0: [0.6, 2.3272727272727276, -0.9872727272727273, 0.5859090909090908, -0.8237878787878787, 1.1314242424242424, 0.6131424242424243]\n",
      "Iteration 1: [1.0301818181818183, 2.116834710743802, -1.3751892561983472, 0.537543227731864, -0.577734254157739, 1.358324168452199, 0.63583241684522]\n",
      "Iteration 2: [1.0867213223140495, 2.0999002166291008, -1.4610971705623732, 0.5551223271160487, -0.5253588151088393, 1.396840629831466, 0.6396840629831466]\n",
      "Iteration 3: [1.1022094557753848, 2.0887041185331694, -1.475111435539595, 0.5611181161460048, -0.5161334026878959, 1.4032751564226138, 0.6403275156422614]\n",
      "Iteration 4: [1.103892698961236, 2.085947901362148, -1.4770544849682161, 0.5624140340214537, -0.5145595170148132, 1.4042371936517284, 0.6404237193651728]\n",
      "Iteration 5: [1.104005687129858, 2.0854281000088437, -1.4772880821184604, 0.5626556750700163, -0.5143188819585031, 1.4043650201803548, 0.6404365020180355]\n",
      "Iteration 6: [1.1040004264245764, 2.0853404835541878, -1.4773099754766013, 0.5626958076585277, -0.5142860168867514, 1.4043794394674336, 0.6404379439467434]\n",
      "Iteration 7: [1.103996043450739, 2.085327149545323, -1.477310744809993, 0.5627018158056072, -0.5142821450288362, 1.404380588831905, 0.6404380588831905]\n",
      "Iteration 8: [1.103994863916531, 2.085325333789974, -1.4773104344733194, 0.5627026182606603, -0.5142818004528367, 1.4043805761397476, 0.6404380576139748]\n",
      "Iteration 9: [1.1039946202736612, 2.0853251210016692, -1.4773103229704236, 0.562702709464311, -0.5142817931395771, 1.404380544024609, 0.6404380544024609]\n",
      "Iteration 10: [1.1039945766942516, 2.085325102302809, -1.477310297369521, 0.5627027168818729, -0.514281799452101, 1.4043805347606821, 0.6404380534760682]\n",
      "Iteration 11: [1.103994569704185, 2.0853251019717316, -1.4773102924836814, 0.5627027168457519, -0.5142818015147649, 1.4043805327897583, 0.6404380532789758]\n",
      "Iteration 12: [1.1039945686939094, 2.0853251023339068, -1.4773102916577434, 0.5627027166521477, -0.5142818019742594, 1.4043805324303686, 0.6404380532430369]\n",
      "Iteration 13: [1.1039945685649393, 2.0853251024500685, -1.4773102915318768, 0.5627027165953048, -0.5142818020604398, 1.4043805323717788, 0.6404380532371778]\n",
      "Converged in 13 iterations\n",
      "Final Solution: [1.1039945685649393, 2.0853251024500685, -1.4773102915318768, 0.5627027165953048, -0.5142818020604398, 1.4043805323717788, 0.6404380532371778]\n",
      "Number of iterations: 13\n",
      "Time taken (seconds): 0.01399993896484375\n"
     ]
    }
   ],
   "source": [
    "# Hard-coded Example Input (7x7 Matrix)\n",
    "A_7x7 = [10.0  -1.0   2.0   0.0   0.0   0.0   0.0;\n",
    "         -1.0  11.0  -1.0   3.0   0.0   0.0   0.0;\n",
    "          2.0  -1.0  10.0  -1.0   0.0   3.0   0.0;\n",
    "          0.0   3.0  -1.0  12.0  -1.0   0.0   0.0;\n",
    "          0.0   0.0   0.0  -1.0   9.0  -2.0   0.0;\n",
    "          0.0   0.0   3.0   0.0  -2.0  10.0  -1.0;\n",
    "          0.0   0.0   0.0   0.0   0.0  -1.0  10.0]\n",
    "\n",
    "b_7x7 = [6.0, 25.0, -11.0, 15.0, -8.0, 10.0, 5.0]\n",
    "\n",
    "x0_7x7 = zeros(7)\n",
    "\n",
    "function main()\n",
    "    start_time = time()\n",
    "    x_sol, iterations = gauss_seidel(A_7x7, b_7x7, x0_7x7)\n",
    "    end_time = time()\n",
    "    duration = end_time - start_time\n",
    "    println(\"Final Solution: \", x_sol)\n",
    "    println(\"Number of iterations: \", iterations)\n",
    "    println(\"Time taken (seconds): \", duration)\n",
    "end\n",
    "\n",
    "main()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0e31829d-697c-4953-9e30-beb078cb0f45",
   "metadata": {},
   "source": [
    "# Hard-coded Example Input (8x8 Matrix)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "da92321e-eae5-43b2-8e07-acb12c106d8e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0: [0.6, 2.3272727272727276, -0.9872727272727273, 0.5859090909090908, -0.8237878787878787, 1.1314242424242424, 0.6131424242424243, 1.051095202020202]\n",
      "Iteration 1: [1.0301818181818183, 2.116834710743802, -1.3751892561983472, 0.537543227731864, -0.577734254157739, 1.358324168452199, 0.7409419370472401, 1.0617451614206033]\n",
      "Iteration 2: [1.0867213223140495, 2.0999002166291008, -1.4610971705623732, 0.5551223271160487, -0.5253588151088393, 1.407351581851668, 0.7469096743272271, 1.0622424728606024]\n",
      "Iteration 3: [1.1022094557753848, 2.0887041185331694, -1.4782647211456557, 0.5608553423454997, -0.5138268326612405, 1.4154050172441714, 0.7477647490104774, 1.0623137290842064]\n",
      "Iteration 4: [1.1045233560824481, 2.0857902370818446, -1.4808616184470067, 0.5623284031371849, -0.5118735069304969, 1.4166602590490505, 0.7478973988133257, 1.0623247832344438]\n",
      "Iteration 5: [1.1047513473975858, 2.0851731381399112, -1.4811981930665226, 0.5626174071319373, -0.511562452752218, 1.4168367072508457, 0.7479161490485289, 1.062326345754044]\n",
      "Iteration 6: [1.1047569524272958, 2.0850642307240874, -1.4812342388751103, 0.5626675513500342, -0.5115176704609193, 1.416858352475202, 0.7479184698229246, 1.0623265391519103]\n",
      "Iteration 7: [1.1047532708474308, 2.0850469434474745, -1.4812367104322959, 0.5626753990636968, -0.5115119884428777, 1.4168604624234056, 0.7479187001575316, 1.062326558346461]\n",
      "Iteration 8: [1.1047520364312067, 2.0850444662552565, -1.4812365594813677, 0.5626765044424987, -0.5115113967456322, 1.416860558511037, 0.7479187116857497, 1.0623265593071458]\n",
      "Iteration 9: [1.1047517585217992, 2.0850441532466304, -1.481236453488758, 0.5626766408354765, -0.51151136023805, 1.4168605351675922, 0.7479187094474737, 1.0623265591206228]\n",
      "Iteration 10: [1.1047517060224146, 2.085044120911566, -1.4812364255800563, 0.5626766542872663, -0.5115113639308387, 1.4168605258325964, 0.7479187084953219, 1.0623265590412767]\n",
      "Iteration 11: [1.1047516972071678, 2.0850441189786646, -1.4812364198646193, 0.5626766549390457, -0.5115113659328624, 1.4168605236223457, 0.7479187082663622, 1.0623265590221969]\n",
      "Iteration 12: [1.1047516958707901, 2.085044119199003, -1.481236418847057, 0.5626766548019225, -0.5115113664392651, 1.4168605231929003, 0.7479187082215097, 1.0623265590184592]\n",
      "Iteration 13: [1.1047516956893118, 2.085044119312408, -1.4812364186842994, 0.5626766547449343, -0.5115113665410295, 1.4168605231192348, 0.7479187082137694, 1.0623265590178141]\n",
      "Converged in 13 iterations\n",
      "Final Solution: [1.1047516956893118, 2.085044119312408, -1.4812364186842994, 0.5626766547449343, -0.5115113665410295, 1.4168605231192348, 0.7479187082137694, 1.0623265590178141]\n",
      "Number of iterations: 13\n",
      "Time taken (seconds): 0.014000177383422852\n"
     ]
    }
   ],
   "source": [
    "# Hard-coded Example Input (8x8 Matrix)\n",
    "A_8x8 = [10.0  -1.0   2.0   0.0   0.0   0.0   0.0   0.0;\n",
    "         -1.0  11.0  -1.0   3.0   0.0   0.0   0.0   0.0;\n",
    "          2.0  -1.0  10.0  -1.0   0.0   3.0   0.0   0.0;\n",
    "          0.0   3.0  -1.0  12.0  -1.0   0.0   0.0   0.0;\n",
    "          0.0   0.0   0.0  -1.0   9.0  -2.0   0.0   0.0;\n",
    "          0.0   0.0   3.0   0.0  -2.0  10.0  -1.0   0.0;\n",
    "          0.0   0.0   0.0   0.0   0.0  -1.0  10.0  -1.0;\n",
    "          0.0   0.0   0.0   0.0   0.0   0.0  -1.0  12.0]\n",
    "\n",
    "b_8x8 = [6.0, 25.0, -11.0, 15.0, -8.0, 10.0, 5.0, 12.0]\n",
    "\n",
    "x0_8x8 = zeros(8)\n",
    "\n",
    "function main()\n",
    "    start_time = time()\n",
    "    x_sol, iterations = gauss_seidel(A_8x8, b_8x8, x0_8x8)\n",
    "    end_time = time()\n",
    "    duration = end_time - start_time\n",
    "    println(\"Final Solution: \", x_sol)\n",
    "    println(\"Number of iterations: \", iterations)\n",
    "    println(\"Time taken (seconds): \", duration)\n",
    "end\n",
    "\n",
    "main()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "665b8739-aa5f-452a-a942-4c5b5a1da71c",
   "metadata": {},
   "source": [
    "# Hard-coded Example Input (9x9 Matrix)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "cf443b80-b5a4-4a30-9b08-af7a441362bb",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0: [0.6, 2.3272727272727276, -0.9872727272727273, 0.5859090909090908, -0.8237878787878787, 1.1314242424242424, 0.6131424242424243, 1.051095202020202, 1.1501564574314573]\n",
      "Iteration 1: [1.0301818181818183, 2.116834710743802, -1.3751892561983472, 0.537543227731864, -0.577734254157739, 1.358324168452199, 0.7409419370472401, 1.157591532873225, 1.1653702189818893]\n",
      "Iteration 2: [1.0867213223140495, 2.0999002166291008, -1.4610971705623732, 0.5551223271160487, -0.5253588151088393, 1.407351581851668, 0.7564943114724894, 1.1601553775378648, 1.1657364825054093]\n",
      "Iteration 3: [1.1022094557753848, 2.0887041185331694, -1.4782647211456557, 0.5608553423454997, -0.5138268326612405, 1.4163634809586976, 0.7576518858496563, 1.160282364029589, 1.1657546234327985]\n",
      "Iteration 4: [1.1045233560824481, 2.0857902370818446, -1.4811491575613644, 0.5623044415443218, -0.5116631773931426, 1.4177773003747465, 0.7578059664404335, 1.1602967158227693, 1.165756673688967]\n",
      "Iteration 5: [1.1048088552204574, 2.0851587611841933, -1.481548640883664, 0.5626093248475511, -0.5113151193781061, 1.4179821650335214, 0.7578278880856291, 1.1602987134812164, 1.165756959068745]\n",
      "Iteration 6: [1.104825604295152, 2.085040817169894, -1.4815947561673422, 0.5626639727454057, -0.5112635219097279, 1.41800851127682, 0.7578307224758036, 1.1602989734620457, 1.1657569962088636]\n",
      "Iteration 7: [1.1048230329504578, 2.085021487140627, -1.4815986139845343, 0.5626727835569881, -0.5112566882099302, 1.4180113188009547, 0.7578310292263001, 1.160299002119597, 1.1657570003027995]\n",
      "Iteration 8: [1.1048218715109697, 2.085018627895952, -1.4815986287971863, 0.5626740666087523, -0.511255921754371, 1.4180115072109118, 0.7578310509330508, 1.1602990042696542, 1.1657570006099505]\n",
      "Iteration 9: [1.1048215885490325, 2.0850182509023263, -1.4815985381219723, 0.5626742322847232, -0.5112558614770504, 1.4180114942344868, 0.7578310498504142, 1.1602990042050305, 1.1657570006007187]\n",
      "Iteration 10: [1.104821532714627, 2.085018208885317, -1.4815985106962675, 0.5626742500975609, -0.5112558623814962, 1.4180114857176225, 0.7578310489922653, 1.1602990041327488, 1.1657570005903928]\n",
      "Iteration 11: [1.1048215230277851, 2.085018205639894, -1.4815985047470983, 0.5626742513293103, -0.5112558641372716, 1.4180114834959017, 0.757831048762865, 1.1602990041127714, 1.1657570005875386]\n",
      "Iteration 12: [1.1048215215134092, 2.0850182057071254, -1.4815985036478088, 0.5626742512577954, -0.5112558646389336, 1.4180114830428425, 0.7578310487155614, 1.1602990041085917, 1.1657570005869415]\n",
      "Iteration 13: [1.1048215213002743, 2.085018205807189, -1.4815985034664092, 0.5626742512060909, -0.5112558647453582, 1.4180114829624073, 0.7578310487070998, 1.1602990041078367, 1.1657570005868336]\n",
      "Converged in 13 iterations\n",
      "Final Solution: [1.1048215213002743, 2.085018205807189, -1.4815985034664092, 0.5626742512060909, -0.5112558647453582, 1.4180114829624073, 0.7578310487070998, 1.1602990041078367, 1.1657570005868336]\n",
      "Number of iterations: 13\n",
      "Time taken (seconds): 0.014999866485595703\n"
     ]
    }
   ],
   "source": [
    "# Hard-coded Example Input (9x9 Matrix)\n",
    "A_9x9 = [10.0  -1.0   2.0   0.0   0.0   0.0   0.0   0.0   0.0;\n",
    "         -1.0  11.0  -1.0   3.0   0.0   0.0   0.0   0.0   0.0;\n",
    "          2.0  -1.0  10.0  -1.0   0.0   3.0   0.0   0.0   0.0;\n",
    "          0.0   3.0  -1.0  12.0  -1.0   0.0   0.0   0.0   0.0;\n",
    "          0.0   0.0   0.0  -1.0   9.0  -2.0   0.0   0.0   0.0;\n",
    "          0.0   0.0   3.0   0.0  -2.0  10.0  -1.0   0.0   0.0;\n",
    "          0.0   0.0   0.0   0.0   0.0  -1.0  10.0  -1.0   0.0;\n",
    "          0.0   0.0   0.0   0.0   0.0   0.0  -1.0  12.0  -1.0;\n",
    "          0.0   0.0   0.0   0.0   0.0   0.0   0.0  -1.0   7.0]\n",
    "\n",
    "b_9x9 = [6.0, 25.0, -11.0, 15.0, -8.0, 10.0, 5.0, 12.0, 7.0]  # Vector b\n",
    "\n",
    "x0_9x9 = zeros(9)  # Initial guess, a vector of zeros (9 elements)\n",
    "\n",
    "# Main program with time duration tracking\n",
    "function main()\n",
    "    # Start timer\n",
    "    start_time = time()\n",
    "\n",
    "    # Call the Gauss-Seidel method with the hardcoded 9x9 matrix inputs\n",
    "    x_sol, iterations = gauss_seidel(A_9x9, b_9x9, x0_9x9)\n",
    "\n",
    "    # End timer and calculate duration\n",
    "    end_time = time()\n",
    "    duration = end_time - start_time\n",
    "\n",
    "    # Output final solution, number of iterations, and time taken\n",
    "    println(\"Final Solution: \", x_sol)\n",
    "    println(\"Number of iterations: \", iterations)\n",
    "    println(\"Time taken (seconds): \", duration)\n",
    "end\n",
    "\n",
    "# Run the main program\n",
    "main()\n"
   ]
  }
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