+ + + + + +
+ + +
+ + +
+
+ +
+The Numerical Recipes codes have been rewritten in Fortran 90/95 and +C/C++ by us. The original source codes are taken from the widely used +software package LAPACK, which follows two other popular packages +developed in the 1970s, namely EISPACK and LINPACK. + +
+ + +
+
+$$ + \mathbf{A} = + \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ + a_{21} & a_{22} & a_{23} & a_{24} \\ + a_{31} & a_{32} & a_{33} & a_{34} \\ + a_{41} & a_{42} & a_{43} & a_{44} + \end{bmatrix}\qquad +\mathbf{I} = + \begin{bmatrix} 1 & 0 & 0 & 0 \\ + 0 & 1 & 0 & 0 \\ + 0 & 0 & 1 & 0 \\ + 0 & 0 & 0 & 1 + \end{bmatrix} +$$ +
+The inverse of a matrix is defined by + +$$ +\mathbf{A}^{-1} \cdot \mathbf{A} = I +$$ +
+ + +
+
+ +
+ +
| Relations | Name | matrix elements |
| \( A = A^{T} \) | symmetric | \( a_{ij} = a_{ji} \) |
| \( A = \left (A^{T} \right )^{-1} \) | real orthogonal | \( \sum_k a_{ik} a_{jk} = \sum_k a_{ki} a_{kj} = \delta_{ij} \) |
| \( A = A^{ * } \) | real matrix | \( a_{ij} = a_{ij}^{ * } \) |
| \( A = A^{\dagger} \) | hermitian | \( a_{ij} = a_{ji}^{ * } \) |
| \( A = \left (A^{\dagger} \right )^{-1} \) | unitary | \( \sum_k a_{ik} a_{jk}^{ * } = \sum_k a_{ki}^{ * } a_{kj} = \delta_{ij} \) |
+
+ + +
+
+For an \( N\times N \) matrix \( \mathbf{A} \) the following properties are all equivalent + +
+ + +
+
+We have an \( N\times N \) matrix A with \( N=100 \) +In C/C++ this would be defined as + +
+ + +
int N = 100;
+ double A[100][100];
+ // initialize all elements to zero
+ for(i=0 ; i < N ; i++) {
+ for(j=0 ; j < N ; j++) {
+ A[i][j] = 0.0;
++Note the way the matrix is organized, row-major order. +
+ + +
+
+We have \( N\times N \) matrices A, B and C and we wish to +evaluate \( A=B+C \). + +$$ +\mathbf{A}= \mathbf{B}\pm\mathbf{C} \Longrightarrow a_{ij} = b_{ij}\pm c_{ij}, +$$ + +In C/C++ this would be coded like + +
+ + +
for(i=0 ; i < N ; i++) {
+ for(j=0 ; j < N ; j++) {
+ a[i][j] = b[i][j]+c[i][j]
++
+ + +
+
+We have \( N\times N \) matrices A, B and C and we wish to +evaluate \( A=BC \). + +$$ +\mathbf{A}=\mathbf{BC} \Longrightarrow a_{ij} = \sum_{k=1}^{n} b_{ik}c_{kj}, +$$ + +In C/C++ this would be coded like + +
+ + +
for(i=0 ; i < N ; i++) {
+ for(j=0 ; j < N ; j++) {
+ for(k=0 ; k < N ; k++) {
+ a[i][j]+=b[i][k]*c[k][j];
++
+ + +
+At least three possibilities in this course + +
+
+
+ + +
int N;
+double ** A;
+A = new double*[N]
+for ( i = 0; i < N; i++)
+ A[i] = new double[N];
++Always free space when you don't need an array anymore. + +
+ + +
for ( i = 0; i < N; i++)
+ delete[] A[i];
+delete[] A;
++
+ + +
+ + +
#include <iostream>
+#include <armadillo>
+
+using namespace std;
+using namespace arma;
+
+int main(int argc, char** argv)
+ {
+ mat A = randu<mat>(5,5);
+ mat B = randu<mat>(5,5);
+
+ cout << A*B << endl;
+
+ return 0;
++ + +
+For people using Ubuntu, Debian, Linux Mint, simply go to the synaptic package manager and install +armadillo from there. +You may have to install Lapack as well. +For Mac and Windows users, follow the instructions from the webpage +http://arma.sourceforge.net. +To compile, use for example (linux/ubuntu) + +
+ + +
c++ -O2 -o program.x program.cpp -larmadillo -llapack -lblas
+
+where the -l option indicates the library you wish to link to.
+
+
+For OS X users you may have to declare the paths to the include files and the libraries as +
+ + +
c++ -O2 -o program.x program.cpp -L/usr/local/lib -I/usr/local/include -larmadillo -llapack -lblas
++ + +
+ + +
#include <iostream>
+#include "armadillo"
+using namespace arma;
+using namespace std;
+
+int main(int argc, char** argv)
+ {
+ // directly specify the matrix size (elements are uninitialised)
+ mat A(2,3);
+ // .n_rows = number of rows (read only)
+ // .n_cols = number of columns (read only)
+ cout << "A.n_rows = " << A.n_rows << endl;
+ cout << "A.n_cols = " << A.n_cols << endl;
+ // directly access an element (indexing starts at 0)
+ A(1,2) = 456.0;
+ A.print("A:");
+ // scalars are treated as a 1x1 matrix,
+ // hence the code below will set A to have a size of 1x1
+ A = 5.0;
+ A.print("A:");
+ // if you want a matrix with all elements set to a particular value
+ // the .fill() member function can be used
+ A.set_size(3,3);
+ A.fill(5.0); A.print("A:");
++ + +
+ + +
mat B;
+
+ // endr indicates "end of row"
+ B << 0.555950 << 0.274690 << 0.540605 << 0.798938 << endr
+ << 0.108929 << 0.830123 << 0.891726 << 0.895283 << endr
+ << 0.948014 << 0.973234 << 0.216504 << 0.883152 << endr
+ << 0.023787 << 0.675382 << 0.231751 << 0.450332 << endr;
+
+ // print to the cout stream
+ // with an optional string before the contents of the matrix
+ B.print("B:");
+
+ // the << operator can also be used to print the matrix
+ // to an arbitrary stream (cout in this case)
+ cout << "B:" << endl << B << endl;
+ // save to disk
+ B.save("B.txt", raw_ascii);
+ // load from disk
+ mat C;
+ C.load("B.txt");
+ C += 2.0 * B;
+ C.print("C:");
++ + +
+ + +
// submatrix types:
+ //
+ // .submat(first_row, first_column, last_row, last_column)
+ // .row(row_number)
+ // .col(column_number)
+ // .cols(first_column, last_column)
+ // .rows(first_row, last_row)
+
+ cout << "C.submat(0,0,3,1) =" << endl;
+ cout << C.submat(0,0,3,1) << endl;
+
+ // generate the identity matrix
+ mat D = eye<mat>(4,4);
+
+ D.submat(0,0,3,1) = C.cols(1,2);
+ D.print("D:");
+
+ // transpose
+ cout << "trans(B) =" << endl;
+ cout << trans(B) << endl;
+
+ // maximum from each column (traverse along rows)
+ cout << "max(B) =" << endl;
+ cout << max(B) << endl;
++ + +
+ + +
// maximum from each row (traverse along columns)
+ cout << "max(B,1) =" << endl;
+ cout << max(B,1) << endl;
+ // maximum value in B
+ cout << "max(max(B)) = " << max(max(B)) << endl;
+ // sum of each column (traverse along rows)
+ cout << "sum(B) =" << endl;
+ cout << sum(B) << endl;
+ // sum of each row (traverse along columns)
+ cout << "sum(B,1) =" << endl;
+ cout << sum(B,1) << endl;
+ // sum of all elements
+ cout << "sum(sum(B)) = " << sum(sum(B)) << endl;
+ cout << "accu(B) = " << accu(B) << endl;
+ // trace = sum along diagonal
+ cout << "trace(B) = " << trace(B) << endl;
+ // random matrix -- values are uniformly distributed in the [0,1] interval
+ mat E = randu<mat>(4,4);
+ E.print("E:");
++ + +
+ + +
// row vectors are treated like a matrix with one row
+ rowvec r;
+ r << 0.59499 << 0.88807 << 0.88532 << 0.19968;
+ r.print("r:");
+
+ // column vectors are treated like a matrix with one column
+ colvec q;
+ q << 0.81114 << 0.06256 << 0.95989 << 0.73628;
+ q.print("q:");
+
+ // dot or inner product
+ cout << "as_scalar(r*q) = " << as_scalar(r*q) << endl;
+
+ // outer product
+ cout << "q*r =" << endl;
+ cout << q*r << endl;
+
+
+ // sum of three matrices (no temporary matrices are created)
+ mat F = B + C + D;
+ F.print("F:");
+
+ return 0;
++ + +
+ + +
#include <iostream>
+#include "armadillo"
+using namespace arma;
+using namespace std;
+
+int main(int argc, char** argv)
+ {
+ cout << "Armadillo version: " << arma_version::as_string() << endl;
+
+ mat A;
+
+ A << 0.165300 << 0.454037 << 0.995795 << 0.124098 << 0.047084 << endr
+ << 0.688782 << 0.036549 << 0.552848 << 0.937664 << 0.866401 << endr
+ << 0.348740 << 0.479388 << 0.506228 << 0.145673 << 0.491547 << endr
+ << 0.148678 << 0.682258 << 0.571154 << 0.874724 << 0.444632 << endr
+ << 0.245726 << 0.595218 << 0.409327 << 0.367827 << 0.385736 << endr;
+
+ A.print("A =");
+
+ // determinant
+ cout << "det(A) = " << det(A) << endl;
++ + +
+ + +
// inverse
+ cout << "inv(A) = " << endl << inv(A) << endl;
+ double k = 1.23;
+
+ mat B = randu<mat>(5,5);
+ mat C = randu<mat>(5,5);
+
+ rowvec r = randu<rowvec>(5);
+ colvec q = randu<colvec>(5);
+
+
+ // examples of some expressions
+ // for which optimised implementations exist
+ // optimised implementation of a trinary expression
+ // that results in a scalar
+ cout << "as_scalar( r*inv(diagmat(B))*q ) = ";
+ cout << as_scalar( r*inv(diagmat(B))*q ) << endl;
+
+ // example of an expression which is optimised
+ // as a call to the dgemm() function in BLAS:
+ cout << "k*trans(B)*C = " << endl << k*trans(B)*C;
+
+ return 0;
++ + +
+We start with the linear set of equations + +$$ + \mathbf{A}\mathbf{x} = \mathbf{w}. +$$ + +We assume also that the matrix \( \mathbf{A} \) is non-singular and that the +matrix elements along the diagonal satisfy \( a_{ii} \ne 0 \). Simple \( 4\times 4 \) example + +$$ +\begin{bmatrix} + a_{11}& a_{12} &a_{13}& a_{14}\\ + a_{21}& a_{22} &a_{23}& a_{24}\\ + a_{31}& a_{32} &a_{33}& a_{34}\\ + a_{41}& a_{42} &a_{43}& a_{44}\\ + \end{bmatrix} \begin{bmatrix} + x_1\\ + x_2\\ + x_3 \\ + x_4 \\ + \end{bmatrix} + =\begin{bmatrix} + w_1\\ + w_2\\ + w_3 \\ + w_4\\ + \end{bmatrix}. +$$ + +
+ + +
+ + +
+The basic idea of Gaussian elimination is to use the first equation to eliminate the first unknown \( x_1 \) +from the remaining \( n-1 \) equations. Then we use the new second equation to eliminate the second unknown +\( x_2 \) from the remaining \( n-2 \) equations. With \( n-1 \) such eliminations +we obtain a so-called upper triangular set of equations of the form + +$$ +\begin{align} + b_{11}x_1 +b_{12}x_2 +b_{13}x_3 + b_{14}x_4=&y_1 \nonumber \\ + b_{22}x_2 + b_{23}x_3 + b_{24}x_4=&y_2 \nonumber \\ +b_{33}x_3 + b_{34}x_4=&y_3 \nonumber \\ +b_{44}x_4=&y_4. \nonumber +\label{eq:gaussbacksub} +\end{align} +$$ + +We can solve this system of equations recursively starting from \( x_n \) (in our case \( x_4 \)) and proceed with +what is called a backward substitution. + +
+ + +
+ + +
+Our actual \( 4\times 4 \) example reads after the first operation + +$$ +\begin{bmatrix} + a_{11}& a_{12} &a_{13}& a_{14}\\ + 0& (a_{22}-\frac{a_{21}a_{12}}{a_{11}}) &(a_{23}-\frac{a_{21}a_{13}}{a_{11}}) & (a_{24}-\frac{a_{21}a_{14}}{a_{11}})\\ +0& (a_{32}-\frac{a_{31}a_{12}}{a_{11}})& (a_{33}-\frac{a_{31}a_{13}}{a_{11}})& (a_{34}-\frac{a_{31}a_{14}}{a_{11}})\\ +0&(a_{42}-\frac{a_{41}a_{12}}{a_{11}}) &(a_{43}-\frac{a_{41}a_{13}}{a_{11}}) & (a_{44}-\frac{a_{41}a_{14}}{a_{11}}) \\ + \end{bmatrix} \begin{bmatrix} + x_1\\ + x_2\\ + x_3 \\ + x_4 \\ + \end{bmatrix} + =\begin{bmatrix} + y_1\\ + w_2^{(2)}\\ + w_3^{(2)} \\ + w_4^{(2)}\\ + \end{bmatrix}, +$$ + +or + +$$ +\begin{align} + b_{11}x_1 +b_{12}x_2 +b_{13}x_3 + b_{14}x_4=&y_1 \nonumber \\ + a^{(2)}_{22}x_2 + a^{(2)}_{23}x_3 + a^{(2)}_{24}x_4=&w^{(2)}_2 \nonumber \\ + a^{(2)}_{32}x_2 + a^{(2)}_{33}x_3 + a^{(2)}_{34}x_4=&w^{(2)}_3 \nonumber \\ + a^{(2)}_{42}x_2 + a^{(2)}_{43}x_3 + a^{(2)}_{44}x_4=&w^{(2)}_4, \nonumber \\ +\label{_auto2} +\end{align} +$$ + +
+ + +
+The new coefficients are + +$$ +\begin{equation} + b_{1k} = a_{1k}^{(1)} \quad k=1,\dots,n, +\label{_auto3} +\end{equation} +$$ + +where each \( a_{1k}^{(1)} \) is equal to the original \( a_{1k} \) element. The other coefficients are + +$$ +\begin{equation} +a_{jk}^{(2)} = a_{jk}^{(1)}-\frac{a_{j1}^{(1)}a_{1k}^{(1)}}{a_{11}^{(1)}} \quad j,k=2,\dots,n, +\label{_auto4} +\end{equation} +$$ + +with a new right-hand side given by + +$$ +\begin{equation} +y_{1}=w_1^{(1)}, \quad w_j^{(2)} =w_j^{(1)}-\frac{a_{j1}^{(1)}w_1^{(1)}}{a_{11}^{(1)}} \quad j=2,\dots,n. +\label{_auto5} +\end{equation} +$$ + +We have also set \( w_1^{(1)}=w_1 \), the original vector element. +We see that the system of unknowns \( x_1,\dots,x_n \) is transformed into an \( (n-1)\times (n-1) \) problem. + +
+ + +
+This step is called forward substitution. +Proceeding with these substitutions, we obtain the +general expressions for the new coefficients + +$$ +\begin{equation} + a_{jk}^{(m+1)} = a_{jk}^{(m)}-\frac{a_{jm}^{(m)}a_{mk}^{(m)}}{a_{mm}^{(m)}} \quad j,k=m+1,\dots,n, +\label{_auto6} +\end{equation} +$$ + +with \( m=1,\dots,n-1 \) and a +right-hand side given by + +$$ +\begin{equation} + w_j^{(m+1)} =w_j^{(m)}-\frac{a_{jm}^{(m)}w_m^{(m)}}{a_{mm}^{(m)}}\quad j=m+1,\dots,n. +\label{_auto7} +\end{equation} +$$ + +This set of \( n-1 \) elimations leads us to an equations which is solved by back substitution. +If the arithmetics is exact and the matrix \( \mathbf{A} \) is not singular, then the computed answer will be exact. + +
+Even though the matrix elements along the diagonal are not zero, +numerically small numbers may appear and subsequent divisions may lead to large numbers, which, if added +to a small number may yield losses of precision. Suppose for example that our first division in \( (a_{22}-a_{21}a_{12}/a_{11}) \) +results in \( -10^{-7} \) and that \( a_{22} \) is one. +one. We are then +adding \( 10^7+1 \). With single precision this results in \( 10^7 \). + +
+ + +
+The LU decomposition method means that we can rewrite +this matrix as the product of two matrices \( \mathbf{L} \) and \( \mathbf{U} \) +where + +$$ + \begin{bmatrix} + a_{11} & a_{12} & a_{13} & a_{14} \\ + a_{21} & a_{22} & a_{23} & a_{24} \\ + a_{31} & a_{32} & a_{33} & a_{34} \\ + a_{41} & a_{42} & a_{43} & a_{44} + \end{bmatrix} + = \begin{bmatrix} + 1 & 0 & 0 & 0 \\ + l_{21} & 1 & 0 & 0 \\ + l_{31} & l_{32} & 1 & 0 \\ + l_{41} & l_{42} & l_{43} & 1 + \end{bmatrix} + \begin{bmatrix} + u_{11} & u_{12} & u_{13} & u_{14} \\ + 0 & u_{22} & u_{23} & u_{24} \\ + 0 & 0 & u_{33} & u_{34} \\ + 0 & 0 & 0 & u_{44} + \end{bmatrix}. +$$ + +
+ + +
+LU decomposition forms the backbone of other algorithms in linear algebra, such as the +solution of linear equations given by + +$$ +\begin{align} + a_{11}x_1 +a_{12}x_2 +a_{13}x_3 + a_{14}x_4=&w_1 \nonumber \\ +a_{21}x_1 + a_{22}x_2 + a_{23}x_3 + a_{24}x_4=&w_2 \nonumber \\ +a_{31}x_1 + a_{32}x_2 + a_{33}x_3 + a_{34}x_4=&w_3 \nonumber \\ +a_{41}x_1 + a_{42}x_2 + a_{43}x_3 + a_{44}x_4=&w_4. \nonumber +\end{align} +$$ + +The above set of equations is conveniently solved by using LU decomposition as an intermediate step. + +
+The matrix \( \mathbf{A}\in \mathbb{R}^{n\times n} \) has an LU factorization if the determinant +is different from zero. If the LU factorization exists and \( \mathbf{A} \) is non-singular, then the LU factorization +is unique and the determinant is given by + +$$ +det\{\mathbf{A}\}=det\{\mathbf{LU}\}= det\{\mathbf{L}\}det\{\mathbf{U}\}=u_{11}u_{22}\dots u_{nn}. +$$ + +
+ + +
+There are at least three main advantages with LU decomposition compared with standard Gaussian elimination: + +
+With the LU decomposition it is rather +simple to solve a system of linear equations + +$$ +\begin{align} + a_{11}x_1 +a_{12}x_2 +a_{13}x_3 + a_{14}x_4=&w_1 \nonumber \\ +a_{21}x_1 + a_{22}x_2 + a_{23}x_3 + a_{24}x_4=&w_2 \nonumber \\ +a_{31}x_1 + a_{32}x_2 + a_{33}x_3 + a_{34}x_4=&w_3 \nonumber \\ +a_{41}x_1 + a_{42}x_2 + a_{43}x_3 + a_{44}x_4=&w_4. \nonumber +\end{align} +$$ + +
+This can be written in matrix form as + +$$ \mathbf{Ax}=\mathbf{w}. $$ + +
+where \( \mathbf{A} \) and \( \mathbf{w} \) are known and we have to solve for +\( \mathbf{x} \). Using the LU dcomposition we write + +$$ \mathbf{A} \mathbf{x} \equiv \mathbf{L} \mathbf{U} \mathbf{x} =\mathbf{w}. $$ + +
+ + +
+The previous equation can be calculated in two steps + +$$ \mathbf{L} \mathbf{y} = \mathbf{w};\qquad \mathbf{Ux}=\mathbf{y}. $$ + +
+To show that this is correct we use to the LU decomposition +to rewrite our system of linear equations as + +$$ \mathbf{LUx}=\mathbf{w}, $$ + +and since the determinat of \( \mathbf{L} \) is equal to 1 (by construction +since the diagonals of \( \mathbf{L} \) equal 1) we can use the inverse of +\( \mathbf{L} \) to obtain + +$$ + \mathbf{Ux}=\mathbf{L^{-1}w}=\mathbf{y}, +$$ + +which yields the intermediate step + +$$ + \mathbf{L^{-1}w}=\mathbf{y} +$$ + +and as soon as we have \( \mathbf{y} \) we can obtain \( \mathbf{x} \) +through \( \mathbf{Ux}=\mathbf{y} \). + +
+ + +
+For our four-dimentional example this takes the form + +$$ +\begin{align} + y_1=&w_1 \nonumber\\ +l_{21}y_1 + y_2=&w_2\nonumber \\ +l_{31}y_1 + l_{32}y_2 + y_3 =&w_3\nonumber \\ +l_{41}y_1 + l_{42}y_2 + l_{43}y_3 + y_4=&w_4. \nonumber +\end{align} +$$ + +
+and + +$$ +\begin{align} + u_{11}x_1 +u_{12}x_2 +u_{13}x_3 + u_{14}x_4=&y_1 \nonumber\\ +u_{22}x_2 + u_{23}x_3 + u_{24}x_4=&y_2\nonumber \\ +u_{33}x_3 + u_{34}x_4=&y_3\nonumber \\ +u_{44}x_4=&y_4 \nonumber +\end{align} +$$ + +
+This example shows the basis for the algorithm +needed to solve the set of \( n \) linear equations. + +
+ + +
+The algorithm goes as follows + +
ludcmp(double a, int n, int indx, double &d). This functions returns the LU decomposed matrix \( \bf A \), its determinant and the vector indx which keeps track of the number of interchanges of rows. If the determinant is zero, the solution is malconditioned.lubksb(double a, int n, int indx, double w) which uses the LU decomposed matrix \( \bf A \) and the vector \( \bf w \) and returns \( \bf x \) in the same place as \( \bf w \). Upon exit the original content in \( \bf w \) is destroyed. If you wish to keep this information, you should make a backup of it in your calling function.+If the inverse exists then + +$$ + \mathbf{A}^{-1}\mathbf{A}=\mathbf{I}, +$$ + +the identity matrix. With an LU decomposed matrix we can rewrite the last equation as + +$$ + \mathbf{LU}\mathbf{A}^{-1}=\mathbf{I}. +$$ + +
+ + +
+If we assume that the first column (that is column 1) of the inverse matrix +can be written as a vector with unknown entries + +$$ + \mathbf{A}_1^{-1}= \begin{bmatrix} + a_{11}^{-1} \\ + a_{21}^{-1} \\ + \dots \\ + a_{n1}^{-1} \\ + \end{bmatrix}, +$$ + +then we have a linear set of equations + +$$ + \mathbf{LU}\begin{bmatrix} + a_{11}^{-1} \\ + a_{21}^{-1} \\ + \dots \\ + a_{n1}^{-1} \\ + \end{bmatrix} =\begin{bmatrix} + 1 \\ + 0 \\ + \dots \\ + 0 \\ + \end{bmatrix}. +$$ + +
+ + +
+In a similar way we can compute the unknow entries of the second column, + +$$ + \mathbf{LU}\begin{bmatrix} + a_{12}^{-1} \\ + a_{22}^{-1} \\ + \dots \\ + a_{n2}^{-1} \\ + \end{bmatrix}=\begin{bmatrix} + 0 \\ + 1 \\ + \dots \\ + 0 \\ + \end{bmatrix}, +$$ + +and continue till we have solved all \( n \) sets of linear equations. + +
+ + +
+ + +
#include <iostream>
+#include "armadillo"
+using namespace arma;
+using namespace std;
+
+int main()
+ {
+ mat A = randu<mat>(5,5);
+ vec b = randu<vec>(5);
+
+ A.print("A =");
+ b.print("b=");
+ // solve Ax = b
+ vec x = solve(A,b);
+ // print x
+ x.print("x=");
+ // find LU decomp of A, if needed, P is the permutation matrix
+ mat L, U;
+ lu(L,U,A);
+ // print l
+ L.print(" L= ");
+ // print U
+ U.print(" U= ");
+ //Check that A = LU
+ (A-L*U).print("Test of LU decomposition");
+ return 0;
+ }
++ + +
+ +
+ + +
+It is a simple method for solving +$$ +\mathbf{A}\mathbf{x}=\mathbf{b}, +$$ + +where \( \mathbf{A} \) is a matrix and \( \mathbf{x} \) and \( \mathbf{b} \) are vectors. The vector \( \mathbf{x} \) is +the unknown. + +
+It is an iterative scheme where we start with a guess for the unknown, and +after \( k+1 \) iterations we have +$$ +\mathbf{x}^{(k+1)}= \mathbf{D}^{-1}(\mathbf{b}-(\mathbf{L}+\mathbf{U})\mathbf{x}^{(k)}), +$$ + +with \( \mathbf{A}=\mathbf{D}+\mathbf{U}+\mathbf{L} \) and +\( \mathbf{D} \) being a diagonal matrix, \( \mathbf{U} \) an upper triangular matrix and \( \mathbf{L} \) a lower triangular +matrix. + +
+If the matrix \( \mathbf{A} \) is positive definite or diagonally dominant, one can show that this method will always converge to the exact solution. +
+ + +
+We can demonstrate Jacobi's method by this \( 4\times 4 \) matrix problem. We assume a guess +for the vector elements \( x_i^{(0)} \), a guess which represents our first iteration. The new +values are obtained by substitution +$$ +\begin{align} + x_1^{(1)} =&(b_1-a_{12}x_2^{(0)} -a_{13}x_3^{(0)} - a_{14}x_4^{(0)})/a_{11} \nonumber \\ + x_2^{(1)} =&(b_2-a_{21}x_1^{(0)} - a_{23}x_3^{(0)} - a_{24}x_4^{(0)})/a_{22} \nonumber \\ + x_3^{(1)} =&(b_3- a_{31}x_1^{(0)} -a_{32}x_2^{(0)} -a_{34}x_4^{(0)})/a_{33} \nonumber \\ + x_4^{(1)}=&(b_4-a_{41}x_1^{(0)} -a_{42}x_2^{(0)} - a_{43}x_3^{(0)})/a_{44}, \nonumber +\end{align} +$$ + +which after \( k+1 \) iterations reads +$$ +\begin{align} + x_1^{(k+1)} =&(b_1-a_{12}x_2^{(k)} -a_{13}x_3^{(k)} - a_{14}x_4^{(k)})/a_{11} \nonumber \\ + x_2^{(k+1)} =&(b_2-a_{21}x_1^{(k)} - a_{23}x_3^{(k)} - a_{24}x_4^{(k)})/a_{22} \nonumber \\ + x_3^{(k+1)} =&(b_3- a_{31}x_1^{(k)} -a_{32}x_2^{(k)} -a_{34}x_4^{(k)})/a_{33} \nonumber \\ + x_4^{(k+1)}=&(b_4-a_{41}x_1^{(k)} -a_{42}x_2^{(k)} - a_{43}x_3^{(k)})/a_{44}, \nonumber +\end{align} +$$ +
+ + +
+We can generalize the above equations to +$$ + x_i^{(k+1)}=(b_i-\sum_{j=1, j\ne i}^{n}a_{ij}x_j^{(k)})/a_{ii} +$$ + +or in an even more compact form as +$$ \mathbf{x}^{(k+1)}= \mathbf{D}^{-1}(\mathbf{b}-(\mathbf{L}+\mathbf{U})\mathbf{x}^{(k)}), +$$ + +with \( \mathbf{A}=\mathbf{D}+\mathbf{U}+\mathbf{L} \) and +\( \mathbf{D} \) being a diagonal matrix, \( \mathbf{U} \) an upper triangular matrix and \( \mathbf{L} \) a lower triangular +matrix. +
+ + +
+Our \( 4\times 4 \) matrix problem +$$ +\begin{align} + x_1^{(k+1)} =&(b_1-a_{12}x_2^{(k)} -a_{13}x_3^{(k)} - a_{14}x_4^{(k)})/a_{11} \nonumber \\ + x_2^{(k+1)} =&(b_2-a_{21}x_1^{(k)} - a_{23}x_3^{(k)} - a_{24}x_4^{(k)})/a_{22} \nonumber \\ + x_3^{(k+1)} =&(b_3- a_{31}x_1^{(k)} -a_{32}x_2^{(k)} -a_{34}x_4^{(k)})/a_{33} \nonumber \\ + x_4^{(k+1)}=&(b_4-a_{41}x_1^{(k)} -a_{42}x_2^{(k)} - a_{43}x_3^{(k)})/a_{44}, \nonumber +\end{align} +$$ + +can be rewritten as +$$ +\begin{align} + x_1^{(k+1)} =&(b_1-a_{12}x_2^{(k)} -a_{13}x_3^{(k)} - a_{14}x_4^{(k)})/a_{11} \nonumber \\ + x_2^{(k+1)} =&(b_2-a_{21}x_1^{(k+1)} - a_{23}x_3^{(k)} - a_{24}x_4^{(k)})/a_{22} \nonumber \\ + x_3^{(k+1)} =&(b_3- a_{31}x_1^{(k+1)} -a_{32}x_2^{(k+1)} -a_{34}x_4^{(k)})/a_{33} \nonumber \\ + x_4^{(k+1)}=&(b_4-a_{41}x_1^{(k+1)} -a_{42}x_2^{(k+1)} - a_{43}x_3^{(k+1)})/a_{44}, \nonumber +\end{align} +$$ + +which allows us to utilize the preceding solution (forward substitution). This improves normally the convergence +behavior and leads to the Gauss-Seidel method! +
+ + +
+We can generalize +$$ +\begin{align} + x_1^{(k+1)} =&(b_1-a_{12}x_2^{(k)} -a_{13}x_3^{(k)} - a_{14}x_4^{(k)})/a_{11} \nonumber \\ + x_2^{(k+1)} =&(b_2-a_{21}x_1^{(k+1)} - a_{23}x_3^{(k)} - a_{24}x_4^{(k)})/a_{22} \nonumber \\ + x_3^{(k+1)} =&(b_3- a_{31}x_1^{(k+1)} -a_{32}x_2^{(k+1)} -a_{34}x_4^{(k)})/a_{33} \nonumber \\ + x_4^{(k+1)}=&(b_4-a_{41}x_1^{(k+1)} -a_{42}x_2^{(k+1)} - a_{43}x_3^{(k+1)})/a_{44}, \nonumber +\end{align} +$$ + +to the following form +$$ + x^{(k+1)}_i = \frac{1}{a_{ii}} \left(b_i - \sum_{j > i}a_{ij}x^{(k)}_j - \sum_{j < i}a_{ij}x^{(k+1)}_j \right),\quad i=1,2,\ldots,n. +$$ + +The procedure is generally continued until the changes made by an iteration are below some tolerance. + +
+The convergence properties of the Jacobi method and the +Gauss-Seidel method are dependent on the matrix \( \mathbf{A} \). These methods converge when +the matrix is symmetric positive-definite, or is strictly or irreducibly diagonally dominant. +Both methods sometimes converge even if these conditions are not satisfied. +
+ + +
+Given a square system of n linear equations with unknown \( \mathbf x \): +$$ + \mathbf{A}\mathbf x = \mathbf b +$$ + +where +$$ + \mathbf{A}=\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix}, \qquad \mathbf{x} = \begin{bmatrix} x_{1} \\ x_2 \\ \vdots \\ x_n \end{bmatrix} , \qquad \mathbf{b} = \begin{bmatrix} b_{1} \\ b_2 \\ \vdots \\ b_n \end{bmatrix}. +$$ +
+ + +
+Then A can be decomposed into a diagonal component D, and strictly lower and upper triangular components L and U: +$$ + \mathbf{A} =\mathbf{D} + \mathbf{L} + \mathbf{U}, +$$ + +where +$$ + D = \begin{bmatrix} a_{11} & 0 & \cdots & 0 \\ 0 & a_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & a_{nn} \end{bmatrix}, \quad L = \begin{bmatrix} 0 & 0 & \cdots & 0 \\ a_{21} & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & \cdots & 0 \end{bmatrix}, \quad U = \begin{bmatrix} 0 & a_{12} & \cdots & a_{1n} \\ 0 & 0 & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & 0 \end{bmatrix}. +$$ + +The system of linear equations may be rewritten as: +$$ + (D+\omega L) \mathbf{x} = \omega \mathbf{b} - [\omega U + (\omega-1) D ] \mathbf{x} +$$ + +for a constant \( \omega > 1 \). +
+ + +
+The method of successive over-relaxation is an iterative technique that solves the left hand side of this expression for \( x \), using previous value for \( x \) on the right hand side. Analytically, this may be written as: +$$ + \mathbf{x}^{(k+1)} = (D+\omega L)^{-1} \big(\omega \mathbf{b} - [\omega U + (\omega-1) D ] \mathbf{x}^{(k)}\big). +$$ + +However, by taking advantage of the triangular form of \( (D+\omega L) \), the elements of \( x^{(k+1)} \) can be computed sequentially using forward substitution: +$$ + x^{(k+1)}_i = (1-\omega)x^{(k)}_i + \frac{\omega}{a_{ii}} \left(b_i - \sum_{j > i} a_{ij}x^{(k)}_j - \sum_{j < i} a_{ij}x^{(k+1)}_j \right),\quad i=1,2,\ldots,n. +$$ + +The choice of relaxation factor is not necessarily easy, and depends upon the properties of the coefficient matrix. For symmetric, positive-definite matrices it can be proven that \( 0 < \omega < 2 \) will lead to convergence, but we are generally interested in faster convergence rather than just convergence. +
+ + + +