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LeastSquaresSolutionQRTallSkinny

Solves overdetermined or underdetermined real / complex linear systems involving an m-by-n matrix A, or its transpose / conjugate-transpose, using a tall skinny QR or short wide LQ factorization of A. It is assumed that A has full rank.

The following options are provided:

1. If trans = 'N' and m >= n:  find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || B - A*X ||.

2. If trans = 'N' and m < n:  find the minimum norm solution of an underdetermined system A * X = B.

3. If trans = 'T' and m >= n:  find the minimum norm solution of an underdetermined system A**T * X = B.

4. If trans = 'T' and m < n:  find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || B - A**T * X ||.

Lapack function GETSLS.

Computing for type matrix<double>

bool  matrix::LeastSquaresSolutionQRTallSkinny(
   ENUM_EQUATIONS_FORM trans,        // form of the system of equations
   matrix&             B,            // right hand side matrix B
   matrix&             X             // solution matrix X
   );
 
bool  matrix::LeastSquaresSolutionQRTallSkinny(
   ENUM_EQUATIONS_FORM trans,        // form of the system of equations
   vector&             B,            // right hand side vector B
   vector&             X             // solution vector X
   );

Computing for type matrix<float>

bool  matrixf::LeastSquaresSolutionQRTallSkinny(
   ENUM_EQUATIONS_FORM trans,        // form of the system of equations
   matrixf&            B,            // right hand side matrix B
   matrixf&            X             // solution matrix X
   );
 
bool  matrixf::LeastSquaresSolutionQRTallSkinny(
   ENUM_EQUATIONS_FORM trans,        // form of the system of equations
   vectorf&            B,            // right hand side vector B
   vectorf&            X             // solution vector X
   );

Computing for type matrix<complex>

bool  matrixc::LeastSquaresSolutionQRTallSkinny(
   ENUM_EQUATIONS_FORM trans,        // form of the system of equations
   matrixc&            B,            // right hand side matrix B
   matrixc&            X             // solution matrix X
   );
 
bool  matrixc::LeastSquaresSolutionQRTallSkinny(
   ENUM_EQUATIONS_FORM trans,        // form of the system of equations
   vectorc&            B,            // right hand side vector B
   vectorc&            X             // solution vector X
   );

Computing for type matrix<complexf>

bool  matrixcf::LeastSquaresSolutionQRTallSkinny(
   ENUM_EQUATIONS_FORM trans,        // form of the system of equations
   matrixcf&           B,            // right hand side matrix B
   matrixcf&           X             // solution matrix X
   );
 
bool  matrixcf::LeastSquaresSolutionQRTallSkinny(
   ENUM_EQUATIONS_FORM trans,        // form of the system of equations
   vectorcf&           B,            // right hand side vector B
   vectorcf&           X             // solution vector X
   );

Parameters

trans

[in]  ENUM_EQUATIONS_FORM enumeration value which specifies the form of the system of equations.

B

[in]  Matrix B whose columns are the right-hand sides for the systems of equations. Vector B contains one column of right-hand side.

X

[out]  Matrix or vector X with solutions of linear least squares problem.

 

Return Value

Return true if successful, otherwise false in case of an error.

Note

If trans='N'. Matrix B must be of size m-by-nrhs, where nrhs is number of right-hand side vectors, matrix X is of size n-by-nrhs. If m>=n, matrix X contains nrhs least squares solution vectors. If m<n, matrix X contains nrhs minimum norm solution vectors.

If trans!='N'. Matrix B must be of size n-by-nrhs, matrix X is of size m-by-nrhs. If m>=n, matrix X contains nrhs minimum norm solution vectors. If m<n, matrix X contains nrhs least squares solution vectors.

ENUM_EQUATIONS_FORM

An enumeration defining which form of the equations' system calculated.

ID

Description

EQUATIONSFORM_N

'N': the linear system involves A (No transpose)

EQUATIONSFORM_T

'T': the linear system involves A**T (Transpose)

EQUATIONSFORM_C

'C': the linear system involves A**H (Conjugate transpose)

In case of real matrices the value EQUATIONSFORM_C assumed as Transpose.

In case of complex matrices the value EQUATIONSFORM_T assumed as Conjugate transpose.