# Systems of Linear Equations

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## Equilibrate Matrices

The equilibration of a matrix is used to transform that matrix to a matrix with a lower condition number. This is usually recommended for using linear equation solvers in which pivoting is used, e.g. a Gaussian elimination routine in which pivoting is performed, a Doolittle routine using pivoting, a Crout routine using pivoting, etc.

Given an equation A x = B where A is an n × n matrix and B and x are n×1 column vectors, this routine calculates n×n non-singular diagonal matrices R and C such that the max norm of the rows and columns of the n×n matrix R A C is bounded by 1. By choosing the diagonal elements of both R and C to be powers of 2, the significands of the corresponding elements of A and R A C agree and the significands of the corresponding elements of B and R B agree. The system A x = B, then is transformed to R A C ( C-1x) = R B. This transformed system is then solved for y = C-1x by using any of the linear equation solvers, in which pivoting is used. After solving for y = C-1x, the value of x is then solved by x = C y.

### Function List

• int Equilibrate_Matrix( double *A, int nrows, int ncols, double r[ ], double c[ ] )

Given the matrix equation A x = B, the routine Equilibrate_Matrix( ) determines the diagonal matrix R with diagonal given by the array r[ ] and the diagonal matrix C with diagonal given by the array c[ ] such that the product R A C equilibrates the matrix A. The array r should be dimensioned at least by nrows in the calling routine and the array c should be dimensioned at least by ncols in the calling routine. Although Equilibrate_Matrix( ) is primarily used to equilibrate an ill-conditioned square matrix, the routine is written to equilibrate an arbitrary nrows×ncols matrix. The original matrix A is replaced with the equilibrated matrix. The function Equilibrate_Matrix( ) returns 0 for success or 1 if an underflow is detected which indicates a loss of significance.

• int Equilibrate_Right_Hand_Side( double B[ ], double r[ ], int n )

After the matrix A is equilibrated, the function Equilibrate_Right_Hand_Side( ) multiplies the right hand side, B, of the linear equation A x = B by the diagonal matrix R. I.e. after calling Equilibrate_Matrix(A,n,n,r,c) and then calling Equilibrate_Right_Hand_Side(B,r,n), the original equation A x = B is transformed to the system (R A C ) C-1 x = R B. The input array r to Equilibrate_Right_Hand_Side( ) is the same array r calculated by Equilibrate_Matrix. The original column vector B is replaced with the product RB. The function Equilibrate_Right_Hand_Side( ) returns 0 for success, 1 if an underflow is detected, 2 if an overflow is detected or 3 if both an underflow and an overflow are detected. If an overflow is detected, then the process is halted immediately and the results are unusable, the column vector B has been destroyed.

• int Unequilibrate_Solution( double x[ ], double c[ ], int n )

After equilibrating the matrix A and the right hand side, B, of the matrix equation
A x = B and then solving the resulting matrix equation, Unequilibrate_Solution( ) transforms the solution of the equilibrated system to a solution of the original system. The input array c to Unequilibrate_Solution( ) is the same array c calculated by Equilibrate_Matrix( ). The input column vector x is the solution to the equilibrated system of linear equations as calculated by a linear equation solver. The original column vector x is replaced with the product Cx. The function Unequilibrate_Solution( ) returns 0 for success, 1 if an underflow is detected, 2 if an overflow is detected or 3 if both an underflow and an overflow are detected. If an overflow is detected, then the results are unusable and the column vector x has been destroyed.

• int Equilibrate_CMatrix( double complex *A, int nrows, int ncols, double r[ ], double c[ ] )

Given the matrix equation A x = B, the routine Equilibrate_CMatrix determines the diagonal matrix R with diagonal given by the array r[ ] and the diagonal matrix C with diagonal given by the array c[ ] such that the product R A C equilibrates the matrix A. The array r should be dimensioned at least by nrows in the calling routine and the array c should be dimensioned at least by ncols in the calling routine. Although Equilibrate_CMatrix is primarily used to equilibrate an ill-conditioned square matrix, the routine is written to equilibrate an arbitrary nrows×ncols matrix. The original matrix A is replaced with the equilibrated matrix. The function Equilibrate_CMatrix( ) returns 0 for success, 1 if an underflow is detected which indicates a loss of significance, 2 if an overflow is detected or 3 if both an underflow and an overflow are detected. If an overflow is detected, then the process is halted immediately and the results are unusable, the matrix A has been destroyed.

• int Equilibrate_CRight_Hand_Side( double complex B[ ], double r[ ], int n )

After the matrix A is equilibrated, the function Equilibrate_CRight_Hand_Side multiplies the right hand side, B, of the linear equation A x = B by the diagonal matrix R. I.e. after calling Equilibrate_CMatrix(A,n,n,r,c) and then calling Equilibrate_CRight_Hand_Side(B,r,n), the original equation A x = B is transformed to the system (R A C ) C-1 x = R B. The input array r to Equilibrate_CRight_Hand_Side is the same array r calculated by Equilibrate_CMatrix. The original column vector B is replaced with the product RB. The function Equilibrate_CRight_Hand_Side( ) returns 0 for success, 1 if an underflow is detected, 2 if an overflow is detected or 3 if both an underflow and an overflow are detected. If an overflow is detected, then the process is halted immediately and the results are unusable, the column vector B has been destroyed.

• int Unequilibrate_CSolution( double complex x[ ], double c[ ], int n )

After equilibrating the matrix A and the right hand side, B, of the matrix equation
A x = B and then solving the resulting matrix equation, Unequilibrate_CSolution( ) transforms the solution of the equilibrated system to a solution of the original system. The input array c to Unequilibrate_CSolution( ) is the same array c calculated by Equilibrate_CMatrix( ). The input column vector x is the solution to the equilibrated system of linear equations as calculated by a linear equation solver. The original column vector x is replaced with the product Cx. The function Unequilibrate_CSolution( ) returns 0 for success, 1 if an underflow is detected, 2 if an overflow is detected or 3 if both an underflow and an overflow are detected. If an overflow is detected, then the results are unusable and the column vector x has been destroyed.

#### C Source Code

• The file, equilibrate_matrix.c, contains the routines Equilibrate_Matrix( ), Equilibrate_Right_Hand_Side( ), and Unequilibrate_Solution( ).

• The file, equilibrate_cmatrix.c, contains the routines Equilibrate_CMatrix( ), Equilibrate_CRight_Hand_Side( ), and Unequilibrate_CSolution( ).

#### C Test Code, Test Results, and Build Shell Script

• The file, testequilibratematrix.c, contains a test program of Equilibrate_Matrix( ), Equilibrate_Right_Hand_Side( ), and Unequilibrate_Solution( ). This program requires the file equilibrate_matrix.c listed above.

• The file, EquilibrateMatrixTests.txt, contains the results of the test program testequilibratematrix.c.

• The file, testequilibratematrix.sh, contains the shell script used to compile, link, and execute the test program testequilibratematrix.c.

• The file, testequilibratecmatrix.c, contains a test program of Equilibrate_CMatrix( ), Equilibrate_CRight_Hand_Side( ), and Unequilibrate_CSolution( ). This program requires the file equilibrate_cmatrix.c listed above.

• The file, EquilibrateCMatrixTests.txt, contains the results of the test program testequilibratecmatrix.c.

• The file, testequilibratecmatrix.sh, contains the shell script used to compile, link, and execute the test program testequilibratecmatrix.c.