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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.
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.
Gaussian elimination has O(n 3) complexity, but introduces division, which results in round-off errors when implemented using floating point numbers. Round-off errors can be avoided if all the numbers are kept as integer fractions instead of floating point. But then the size of each element grows in size exponentially with the number of rows.
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.
Gaussian algorithm may refer to: Gaussian elimination for solving systems of linear equations; Gauss's algorithm for Determination of the day of the week; Gauss's method for preliminary orbit determination; Gauss's Easter algorithm; Gauss separation algorithm
Example with infinitely many solutions: 3x + 3y = 3, 2x + 2y = 2, x + y = 1. Example with no solution: 3 x + 3 y + 3 z = 3, 2 x + 2 y + 2 z = 2, x + y + z = 1, x + y + z = 4. These results may be easier to understand by putting the augmented matrix of the coefficients of the system in row echelon form by using Gaussian elimination .