En álgebra lineal, un descomposición matricial o factorización matricial es una factorización de una matriz en un producto de matrices. Hay muchas descomposiciones matriciales diferentes. Uno de ellos es Descomposición de Cholesky .
El Descomposición de Cholesky o factorización de Cholesky es una descomposición de una matriz hermitiana definida positiva en el producto de una matriz triangular inferior y su transpuesta conjugada. La descomposición de Cholesky es aproximadamente dos veces más eficiente que la Descomposición LU para resolver sistemas de ecuaciones lineales.
La descomposición de Cholesky de una matriz hermitiana definida positiva A es una descomposición de la forma A = [L][L]t , dónde l es una matriz triangular inferior con entradas diagonales reales y positivas, y lt denota la transpuesta conjugada de L. Cada matriz hermitiana positiva-definida (y por tanto también cada matriz simétrica positiva-definida de valor real) tiene una descomposición de Cholesky única.
![left[egin{array}{lll} A_{00} & A_{01} & A_{02} A_{10} & A_{11} & A_{12} A_{20} & A_{ 21} & A_{22} end{array}
ight]=left[egin{array}{lll} L_{00} & 0 & 0 L_{10} & L_{11} & 0 L_{20} y L_{21} y L_{22} end{array}
ight]left[egin{array}{ccc} L_{00} & L_{10} & L_{20} 0 & L_{11} & L_{21} 0 & 0 & L_{22} end{array}
ight]](http://techcodeview.com/img/it-problems-solutions/50/cholesky-decomposition.webp)
Cada matriz A definida positiva y simétrica se puede descomponer en un producto de una matriz triangular inferior única L y su transpuesta: A = L Lt
Las siguientes fórmulas se obtienen resolviendo la matriz triangular inferior anterior y su transpuesta. Estas son las bases del algoritmo de descomposición de Cholesky:
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Ejemplo :
Input :>
![left[egin{array}{ccc} 4 y 12 y -16 12 y 37 y -43 -16 y -43 y 98 end{array}
ight]](http://techcodeview.com/img/it-problems-solutions/50/cholesky-decomposition-3.webp)
Output :>
![left[egin{array}{ccc} 2 y 0 y 0 6 y 1 y 0 -8 y 5 y 3 end{array}
ight]left[egin{array}{ccc} 2 y 6 y -8 0 y 1 y 5 0 y 0 y 3 end{array}
ight]](http://techcodeview.com/img/it-problems-solutions/50/cholesky-decomposition-4.webp)
A continuación se muestra la implementación de la descomposición de Cholesky.
C++
// CPP program to decompose a matrix using> // Cholesky Decomposition> #include> using> namespace> std;> const> int> MAX = 100;> void> Cholesky_Decomposition(>int> matrix[][MAX],> >int> n)> {> >int> lower[n][n];> >memset>(lower, 0,>sizeof>(lower));> >// Decomposing a matrix into Lower Triangular> >for> (>int> i = 0; i for (int j = 0; j <= i; j++) { int sum = 0; if (j == i) // summation for diagonals { for (int k = 0; k sum += pow(lower[j][k], 2); lower[j][j] = sqrt(matrix[j][j] - sum); } else { // Evaluating L(i, j) using L(j, j) for (int k = 0; k sum += (lower[i][k] * lower[j][k]); lower[i][j] = (matrix[i][j] - sum) / lower[j][j]; } } } // Displaying Lower Triangular and its Transpose cout << setw(6) << ' Lower Triangular' << setw(30) << 'Transpose' << endl; for (int i = 0; i // Lower Triangular for (int j = 0; j cout << setw(6) << lower[i][j] << ' '; cout << ' '; // Transpose of Lower Triangular for (int j = 0; j cout << setw(6) << lower[j][i] << ' '; cout << endl; } } // Driver Code int main() { int n = 3; int matrix[][MAX] = { { 4, 12, -16 }, { 12, 37, -43 }, { -16, -43, 98 } }; Cholesky_Decomposition(matrix, n); return 0; }> |
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Java
// Java program to decompose> // a matrix using Cholesky> // Decomposition> class> GFG {> >// static int MAX = 100;> >static> void> Cholesky_Decomposition(>int>[][] matrix,> >int> n)> >{> >int>[][] lower =>new> int>[n][n];> >// Decomposing a matrix> >// into Lower Triangular> >for> (>int> i =>0>; i for (int j = 0; j <= i; j++) { int sum = 0; // summation for diagonals if (j == i) { for (int k = 0; k sum += (int)Math.pow(lower[j][k], 2); lower[j][j] = (int)Math.sqrt( matrix[j][j] - sum); } else { // Evaluating L(i, j) // using L(j, j) for (int k = 0; k sum += (lower[i][k] * lower[j][k]); lower[i][j] = (matrix[i][j] - sum) / lower[j][j]; } } } // Displaying Lower // Triangular and its Transpose System.out.println(' Lower Triangular Transpose'); for (int i = 0; i // Lower Triangular for (int j = 0; j System.out.print(lower[i][j] + ' '); System.out.print(''); // Transpose of // Lower Triangular for (int j = 0; j System.out.print(lower[j][i] + ' '); System.out.println(); } } // Driver Code public static void main(String[] args) { int n = 3; int[][] matrix = new int[][] { { 4, 12, -16 }, { 12, 37, -43 }, { -16, -43, 98 } }; Cholesky_Decomposition(matrix, n); } } // This code is contributed by mits> |
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Python3
# Python3 program to decompose> # a matrix using Cholesky> # Decomposition> import> math> MAX> => 100>;> def> Cholesky_Decomposition(matrix, n):> >lower>=> [[>0> for> x>in> range>(n>+> 1>)]> >for> y>in> range>(n>+> 1>)];> ># Decomposing a matrix> ># into Lower Triangular> >for> i>in> range>(n):> >for> j>in> range>(i>+> 1>):> >sum1>=> 0>;> ># summation for diagonals> >if> (j>=>=> i):> >for> k>in> range>(j):> >sum1>+>=> pow>(lower[j][k],>2>);> >lower[j][j]>=> int>(math.sqrt(matrix[j][j]>-> sum1));> >else>:> > ># Evaluating L(i, j)> ># using L(j, j)> >for> k>in> range>(j):> >sum1>+>=> (lower[i][k]>*>lower[j][k]);> >if>(lower[j][j]>>0>):> >lower[i][j]>=> int>((matrix[i][j]>-> sum1)>/> >lower[j][j]);> ># Displaying Lower Triangular> ># and its Transpose> >print>(>'Lower Triangular Transpose'>);> >for> i>in> range>(n):> > ># Lower Triangular> >for> j>in> range>(n):> >print>(lower[i][j], end>=> ' '>);> >print>('>', end = '> ');> > ># Transpose of> ># Lower Triangular> >for> j>in> range>(n):> >print>(lower[j][i], end>=> ' '>);> >print>('');> # Driver Code> n>=> 3>;> matrix>=> [[>4>,>12>,>->16>],> >[>12>,>37>,>->43>],> >[>->16>,>->43>,>98>]];> Cholesky_Decomposition(matrix, n);> # This code is contributed by mits> |
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C#
// C# program to decompose> // a matrix using Cholesky> // Decomposition> using> System;> class> GFG {> >// static int MAX = 100;> >static> void> Cholesky_Decomposition(>int>[, ] matrix,> >int> n)> >{> >int>[, ] lower =>new> int>[n, n];> >// Decomposing a matrix> >// into Lower Triangular> >for> (>int> i = 0; i for (int j = 0; j <= i; j++) { int sum = 0; // summation for diagonals if (j == i) { for (int k = 0; k sum += (int)Math.Pow(lower[j, k], 2); lower[j, j] = (int)Math.Sqrt( matrix[j, j] - sum); } else { // Evaluating L(i, j) // using L(j, j) for (int k = 0; k sum += (lower[i, k] * lower[j, k]); lower[i, j] = (matrix[i, j] - sum) / lower[j, j]; } } } // Displaying Lower // Triangular and its Transpose Console.WriteLine( ' Lower Triangular Transpose'); for (int i = 0; i // Lower Triangular for (int j = 0; j Console.Write(lower[i, j] + ' '); Console.Write(''); // Transpose of // Lower Triangular for (int j = 0; j Console.Write(lower[j, i] + ' '); Console.WriteLine(); } } // Driver Code static int Main() { int n = 3; int[, ] matrix = { { 4, 12, -16 }, { 12, 37, -43 }, { -16, -43, 98 } }; Cholesky_Decomposition(matrix, n); return 0; } } // This code is contributed by mits> |
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PHP
// PHP program to decompose // a matrix using Cholesky // Decomposition $MAX = 100; function Cholesky_Decomposition($matrix, $n) { $lower; for ($i = 0; $i <= $n; $i++) for ($j = 0; $j <= $n; $j++) $lower[$i][$j] = 0; // Decomposing a matrix // into Lower Triangular for ($i = 0; $i <$n; $i++) { for ($j = 0; $j <= $i; $j++) { $sum = 0; // summation for diagonals if ($j == $i) { for ($k = 0; $k <$j; $k++) $sum += pow($lower[$j][$k], 2); $lower[$j][$j] = sqrt($matrix[$j][$j] - $sum); } else { // Evaluating L(i, j) // using L(j, j) for ($k = 0; $k <$j; $k++) $sum += ($lower[$i][$k] * $lower[$j][$k]); $lower[$i][$j] = ($matrix[$i][$j] - $sum) / $lower[$j][$j]; } } } // Displaying Lower Triangular // and its Transpose echo ' Lower Triangular' . str_pad('Transpose', 30, ' ', STR_PAD_BOTH) . '
'; for ($i = 0; $i <$n; $i++) { // Lower Triangular for ($j = 0; $j <$n; $j++) echo str_pad($lower[$i][$j], 6, ' ', STR_PAD_BOTH).' '; echo ' '; // Transpose of // Lower Triangular for ($j = 0; $j <$n; $j++) echo str_pad($lower[$j][$i], 6, ' ', STR_PAD_BOTH).' '; echo '
'; } } // Driver Code $n = 3; $matrix = array(array(4, 12, -16), array(12, 37, -43), array(-16, -43, 98)); Cholesky_Decomposition($matrix, $n); // This code is contributed by vt_m. ?>> |
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JavaScript
> // javascript program to decompose> // a matrix using Cholesky> // Decomposition> >// function MAX = 100;> function> Cholesky_Decomposition(matrix,n)> {> >var> lower = Array(n).fill(0).map(x =>Matriz(n).fill(0));> >// Decomposing a matrix> >// into Lower Triangular> >for> (>var> i = 0; i for (var j = 0; j <= i; j++) { var sum = 0; // summation for diagonals if (j == i) { for (var k = 0; k sum += parseInt(Math.pow(lower[j][k], 2)); lower[j][j] = parseInt(Math.sqrt( matrix[j][j] - sum)); } else { // Evaluating L(i, j) // using L(j, j) for (var k = 0; k sum += (lower[i][k] * lower[j][k]); lower[i][j] = (matrix[i][j] - sum) / lower[j][j]; } } } // Displaying Lower // Triangular and its Transpose document.write(' Lower Triangular Transpose '); for (var i = 0; i // Lower Triangular for (var j = 0; j document.write(lower[i][j] + ' '); // Transpose of // Lower Triangular for (var j = 0; j document.write(lower[j][i] + ' '); document.write(' '); } } // Driver Code var n = 3; var matrix = [[ 4, 12, -16 ], [ 12, 37, -43 ], [ -16, -43, 98 ] ]; Cholesky_Decomposition(matrix, n); // This code contributed by Princi Singh> |
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Producción
Lower Triangular Transpose 2 0 0 2 6 -8 6 1 0 0 1 5 -8 5 3 0 0 3>
Complejidad del tiempo: O(n^3)
Espacio Auxiliar: O(n^2)