vsip_dllsqsol_p - Solve Linear Least Squares Problem

Core

This manual is preliminary and incomplete.
While our Core implementation implements all functions given in the standard we are still working on completing this documentation.

Please refer to the VSIPL standard for a complete function reference of the Core profile until we have completed work on this documentation.

[next] [prev] [prev-tail] [tail] [up]

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:

min∥Ax -b∥2 x

where $A$ is an $M ×N$ matrix with $M ≥ N$ , $b$ is an $M$ -dimensional vector, and $x$ is the $N$ -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 $A$ is rank-deficient.

Parameters

const vsip_dmview_p* A: Pointer to the $M × N$ matrix view of coefficients.
const vsip_dvview_p* b: Pointer to the $M$ -dimensional vector view containing the right-hand side.
const vsip_dvview_p* x: Pointer to the $N$ -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 $M$ in matrix $A$ must be greater than or equal to the number of columns $N$ .
The solution $x$ minimizes the 2-norm of the residual vector $Ax -b$ .
If $A$ has full column rank, the solution is unique. If $A$ is rank-deficient, the function returns a basic solution with at most $rank(A)$ non-zero components.
This function is particularly useful for overdetermined systems where there is no exact solution, but a best-fit solution is desired.

[next] [prev] [prev-tail] [front] [up]