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The Jacobi Method

The Jacobi method is easily derived by examining each of the n equations in the linear system in isolation. If in the th equation

we solve for the value of while assuming the other entries of remain fixed, we obtain

This suggests an iterative method defined by

which is the Jacobi method. Note that the order in which the equations are examined is irrelevant, since the Jacobi method treats them independently. For this reason, the Jacobi method is also known as the method of simultaneous displacements, since the updates could in principle be done simultaneously.

In matrix terms, the definition of the Jacobi method in () can be expressed as

where the matrices , and represent the diagonal, the strictly lower-triangular, and the strictly upper-triangular parts of , respectively.

The pseudocode for the Jacobi method is given in Figure . Note that an auxiliary storage vector, is used in the algorithm. It is not possible to update the vector in place, since values from are needed throughout the computation of .