LU decomposition
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Every square matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} can be decomposed into a product of a lower triangular matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} and a upper triangular matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} , as described in LU decomposition.
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = LU}
It is a modified form of Gaussian elimination. While the Cholesky decomposition only works for symmetric, positive definite matrices, the more general LU decomposition works for any square matrix.
There are several algorithms for calculating L and U. To derive Crout's algorithm for a 3x3 example, we have to solve the following system:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \begin{pmatrix} a_{11} & a_{21} & a_{31}\\ a_{21} & a_{22} & a_{32}\\ a_{31} & a_{32} & a_{33}\\ \end{pmatrix} = \begin{pmatrix} l_{11} & 0 & 0 \\ l_{21} & l_{22} & 0 \\ l_{31} & l_{32} & l_{33}\\ \end{pmatrix} \begin{pmatrix} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{pmatrix} = LU }
We now would have to solve 9 equations with 12 unknowns. To make the system uniquely solvable, usually the diagonal elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} are set to 1
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_{11}=1}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_{22}=1}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_{33}=1}
so we get a solvable system of 9 unknowns and 9 equations.
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \begin{pmatrix} a_{11} & a_{21} & a_{31}\\ a_{21} & a_{22} & a_{32}\\ a_{31} & a_{32} & a_{33}\\ \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1\\ \end{pmatrix} \begin{pmatrix} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{pmatrix} = \begin{pmatrix} u_{11} & u_{12} & u_{13} \\ u_{11}l_{21} & u_{12}l_{21}+u_{22} & u_{13}l_{21}+u_{23} \\ u_{11}l_{31} & u_{12}l_{31}+u_{22}l_{32} & u_{13}l_{31} + u_{23}l_{32}+u_{33} \end{pmatrix} = LU }
Solving for the other Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} , we get the following equations:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{11}=a_{11}}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{12}=a_{12}}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{13}=a_{13}}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{22}=a_{22} - u_{12}l_{21}}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{23}=a_{23} - u_{13}l_{21}}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{33}=a_{33} - (u_{13}l_{31} + u_{23}l_{32})}
and for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l} :
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_{21}=\frac{1}{u_{11}} a_{21}}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_{31}=\frac{1}{u_{11}} a_{31}}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_{32}=\frac{1}{u_{22}} (a_{32} - u_{12}l_{31}}
We see that there is a calculation pattern, which can be expressed as the following formulas, first for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{ij} = a_{ij} - \sum_{k=1}^{i-1} u_{kj}l_{ik}}
and then for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_{ij} = \frac{1}{u_{jj}} (a_{ij} - \sum_{k=1}^{j-1} u_{kj}l_{ik})}
We see in the second formula that to get the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_{ij}} below the diagonal, we have to divide by the diagonal element (pivot) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{ij}} , so we get problems when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{ij}} is either 0 or very small, which leads to numerical instability.
The solution to this problem is pivoting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , which means rearranging the rows of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , prior to the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle LU} decomposition, in a way that the largest element of each column gets onto the diagonal of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} . Rearranging the columns means to multiply Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} by a permutation matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} :
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle PA \Rightarrow A'}
Example:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} \Rightarrow \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} }
The decomposition algorithm is then applied on the rearranged matrix so that
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle PA = LU}
Task description
The task is to implement a routine which will take a square nxn matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and return a lower triangular matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} , a upper triangular matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} and a permutation matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} , so that the above equation is fullfilled. You should then test it on the following two examples and include your output.
Example 1:
A 1 3 5 2 4 7 1 1 0 L 1.00000 0.00000 0.00000 0.50000 1.00000 0.00000 0.50000 -1.00000 1.00000 U 2.00000 4.00000 7.00000 0.00000 1.00000 1.50000 0.00000 0.00000 -2.00000 P 0 1 0 1 0 0 0 0 1
Example 2:
A 11 9 24 2 1 5 2 6 3 17 18 1 2 5 7 1 L 1.00000 0.00000 0.00000 0.00000 0.27273 1.00000 0.00000 0.00000 0.09091 0.28750 1.00000 0.00000 0.18182 0.23125 0.00360 1.00000 U 11.00000 9.00000 24.00000 2.00000 0.00000 14.54545 11.45455 0.45455 0.00000 0.00000 -3.47500 5.68750 0.00000 0.00000 0.00000 0.51079 P 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1
Common Lisp
Uses the routine (mmul A B) from Matrix multiplication.
<lang lisp>;; Creates a nxn identity matrix. (defun eye (n)
(let ((I (make-array `(,n ,n) :initial-element 0)))
(loop for j from 0 to (- n 1) do
(setf (aref I j j) 1))
I))
- Swap two rows l and k of a mxn matrix A, which is a 2D array.
(defun swap-rows (A l k)
(let* ((n (cadr (array-dimensions A)))
(row (make-array n :initial-element 0)))
(loop for j from 0 to (- n 1) do
(setf (aref row j) (aref A l j))
(setf (aref A l j) (aref A k j))
(setf (aref A k j) (aref row j)))))
- Creates the pivoting matrix for A.
(defun pivotize (A)
(let* ((n (car (array-dimensions A)))
(P (eye n)))
(loop for j from 0 to (- n 1) do
(let ((max (aref A j j))
(row j))
(loop for i from j to (- n 1) do
(if (> (aref A i j) max)
(setq max (aref A i j)
row i)))
(if (not (= j row))
(swap-rows P j row))))
;; Return P. P))
- Decomposes a square matrix A by PA=LU and returns L, U and P.
(defun lu (A)
(let* ((n (car (array-dimensions A)))
(L (make-array `(,n ,n) :initial-element 0))
(U (make-array `(,n ,n) :initial-element 0))
(P (pivotize A))
(A (mmul P A)))
(loop for j from 0 to (- n 1) do
(setf (aref L j j) 1)
(loop for i from 0 to j do
(setf (aref U i j)
(- (aref A i j)
(loop for k from 0 to (- i 1)
sum (* (aref U k j)
(aref L i k))))))
(loop for i from j to (- n 1) do
(setf (aref L i j)
(/ (- (aref A i j)
(loop for k from 0 to (- j 1)
sum (* (aref U k j)
(aref L i k))))
(aref U j j)))))
;; Return L, U and P. (values L U P)))</lang>
Example 1:
<lang lisp>(setf g (make-array '(3 3) :initial-contents '((1 3 5) (2 4 7)(1 1 0))))
- 2A((1 3 5) (2 4 7) (1 1 0))
(lu g)
- 2A((1 0 0) (1/2 1 0) (1/2 -1 1))
- 2A((2 4 7) (0 1 3/2) (0 0 -2))
- 2A((0 1 0) (1 0 0) (0 0 1))</lang>
Example 2:
<lang lisp>(setf h (make-array '(4 4) :initial-contents '((11 9 24 2)(1 5 2 6)(3 17 18 1)(2 5 7 1))))
- 2A((11 9 24 2) (1 5 2 6) (3 17 18 1) (2 5 7 1))
(lup h)
- 2A((1 0 0 0) (3/11 1 0 0) (1/11 23/80 1 0) (2/11 37/160 1/278 1))
- 2A((11 9 24 2) (0 160/11 126/11 5/11) (0 0 -139/40 91/16) (0 0 0 71/139))
- 2A((1 0 0 0) (0 0 1 0) (0 1 0 0) (0 0 0 1))</lang>