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A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If A is an n × n square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the n × n identity matrix is augmented to the right of A, forming an n × 2n block matrix [A | I].
Before Gauss many mathematicians in Eurasia were performing and perfecting it yet as the method became relegated to school grade, few of them left any detailed descriptions. Thus the name Gaussian elimination is only a convenient abbreviation of a complex history. The Polish astronomer Tadeusz Banachiewicz introduced the LU decomposition in ...
Simplified forms of Gaussian elimination have been developed for these situations. [ 6 ] The textbook Numerical Mathematics by Alfio Quarteroni , Sacco and Saleri, lists a modified version of the algorithm which avoids some of the divisions (using instead multiplications), which is beneficial on some computer architectures.
Otherwise, the Bareiss algorithm may be viewed as a variant of Gaussian elimination and needs roughly the same number of arithmetic operations. It follows that, for an n × n matrix of maximum (absolute) value 2 L for each entry, the Bareiss algorithm runs in O( n 3 ) elementary operations with an O( n n /2 2 nL ) bound on the absolute value of ...
The reduced row echelon form of a matrix is unique and does not depend on the sequence of elementary row operations used to obtain it. The variant of Gaussian elimination that transforms a matrix to reduced row echelon form is sometimes called Gauss–Jordan elimination. A matrix is in column echelon form if its transpose is in row echelon form.
The Schur complement arises when performing a block Gaussian elimination on the matrix M.In order to eliminate the elements below the block diagonal, one multiplies the matrix M by a block lower triangular matrix on the right as follows: = [] [] [] = [], where I p denotes a p×p identity matrix.
The Gaussian elimination is a similar algorithm; it transforms any matrix to row echelon form. [51] Both methods proceed by multiplying the matrix by suitable elementary matrices , which correspond to permuting rows or columns and adding multiples of one row to another row.
The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, [5] as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method.