6.2.3 vsip_dllsqsol_p - Solve Linear Least Squares Problem

int vsip_llsqsol_f(const vsip_mview_f* A, const vsip_vview_f* b, const vsip_vview_f* x); 
int vsip_cllsqsol_f(const vsip_cmview_f* A, const vsip_cvview_f* b, const vsip_cvview_f* x);

Description

This function solves the linear least squares problem:

where is an matrix with , is an -dimensional vector, and is the -dimensional solution vector that minimizes the Euclidean norm of the residual vector.

The function uses QR decomposition with column pivoting to solve the least squares problem, which provides a numerically stable solution even when matrix is rank-deficient.

Parameters

• const vsip_dmview_p* A: Pointer to the matrix view of coefficients.

• const vsip_dvview_p* b: Pointer to the -dimensional vector view containing the right-hand side.

• const vsip_dvview_p* x: Pointer to the -dimensional vector view where the least squares solution will be stored.

Return Value

• Returns 0 on success.

• Returns a non-zero value on error (e.g., if matrix dimensions are incompatible or memory allocation fails).

Example

vsip_mview_f *A;             // M×N coefficient matrix 
vsip_vview_f *b;             // M-dimensional right-hand side vector 
vsip_vview_f *x;             // N-dimensional solution vector 
int result; 
 
// Assuming A, b, and x have been properly initialized with appropriate dimensions 
result = vsip_llsqsol_f(A, b, x); 
 
if (result != 0) { 
    // Handle error 
}

Notes

• The number of rows in matrix must be greater than or equal to the number of columns .

• The solution minimizes the 2-norm of the residual vector .

• If has full column rank, the solution is unique. If is rank-deficient, the function returns a basic solution with at most non-zero components.

• This function is particularly useful for overdetermined systems where there is no exact solution, but a best-fit solution is desired.