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MQL5 リファレンス行列とベクトルのメソッドOpenBLASLeast SquaresLeastSquaresSolutionDC 

LeastSquaresSolutionDC

Computes the minimum-norm solution to a real or complex linear least squares problem minimize 2-norm(| b - A*x |) using the singular value decomposition (SVD) of A. A is an m-by-n matrix which may be rank-deficient. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. LAPACK function GELSD.

Computing for type matrix<double>

bool  matrix::LeastSquaresSolutionDC(
  matrix&       B,               // right hand side matrix B
  double       rcond,           // condition to determine matrix A rank
  matrix&         X,               // solution matrix X
  matrix&         residuals,       // matrix with residual sums of squares
  vector&       singular_values,  // vector of singular values
  long           rank             // effective rank of A
  );
 
bool  matrix::LeastSquaresSolutionDC(
  vector&       B,               // right hand side vector B
  double       rcond,           // condition to determine matrix A rank
  vector&         X,               // solution vector X
  vector&         residuals,       // vector with residual sums of squares
  vector&       singular_values,  // vector of singular values
  long           rank             // effective rank of A
  );

Computing for type matrix<float>

bool  matrixf::LeastSquaresSolutionDC(
  matrixf&     B,               // right hand side matrix B
  float         rcond,           // condition to determine matrix A rank
  matrixf&       X,               // solution matrix X
  matrixf&       residuals,       // matrix with residual sums of squares
  vectorf&     singular_values,  // vector of singular values
  long           rank             // effective rank of A
  );
 
bool  matrixf::LeastSquaresSolutionDC(
  vectorf&       B,               // right hand side vector B
  float         rcond,           // condition to determine matrix A rank
  vectorf&       X,               // solution vector X
  vectorf&       residuals,       // vector with residual sums of squares
  vectorf&     singular_values,  // vector of singular values
  long           rank             // effective rank of A
  );

Computing for type matrix<complex>

bool  matrixc::LeastSquaresSolutionDC(
  matrixc&     B,               // right hand side matrix B
  double       rcond,           // condition to determine matrix A rank
  matrixc&       X,               // solution matrix X
  matrixc&       residuals,       // matrix with residual sums of squares
  vector&       singular_values,  // vector of singular values
  long           rank             // effective rank of A
  );
 
bool  matrixc::LeastSquaresSolutionDC(
  vectorc&     B,               // right hand side vector B
  double       rcond,           // condition to determine matrix A rank
  vectorc&       X,               // solution vector X
  vectorc&       residuals,       // vector with residual sums of squares
  vector&       singular_values,  // vector of singular values
  long           rank             // effective rank of A
  );

Computing for type matrix<complexf>

bool  matrixcf::LeastSquaresSolutionDC(
  matrixcf&     B,               // right hand side matrix B
  float         rcond,           // condition to determine matrix A rank
  matrixcf&       X,               // solution matrix X
  matrixcf&       residuals,       // matrix with residual sums of squares
  vectorf&     singular_values,  // vector of singular values
  long           rank             // effective rank of A
  );
 
bool  matrixcf::LeastSquaresSolutionDC(
  vectorcf&     B,               // right hand side vector B
  float         rcond,           // condition to determine matrix A rank
  vectorcf&       X,               // solution vector X
  vectorcf&       residuals,       // vector with residual sums of squares
  vectorf&     singular_values,  // vector of singular values
  long           rank             // effective rank of A
  );

Parameters

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.

rcond

[in] rcond is used to determine the effective rank of A. Singular values S(i) <= rcond*S(1) are treated as zero. If rcond < 0, machine precision is used instead.

X

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

residuals

[out]  Matrix or vector with the residual sum of squares for the solution in each column is given by the sum of squares of elements in that column. If m<=n, then residuals matrix(vector) is empty.

singular_values

[out] Vector of computed singular values.

rank

[out]  The effective rank of A, i.e., the number of singular values which are greater than rcond*S(1).

 

Return Value

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

Note

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.

The condition number of A in the 2-norm = S(1)/S(min(m,n)).

The problem is solved in three steps:

Step 1. Reduce the coefficient matrix A to bidiagonal form with Householder transformations, reducing the original problem into a "bidiagonal least squares problem" (BLS).

Step 2. Solve the BLS using a divide and conquer approach.

Step 3. Apply back all the Householder transformations to solve the original least squares problem.