Prototype

There are two prototypes of gelsd available, please see below.

gelsd( MatrixA& a, MatrixB& b, VectorS& s, const Scalar >,
        int_t& rank );

gelsd( const MatrixA& a, MatrixB& b, VectorS& s, const Scalar >,
        int_t& rank );

Description

gelsd (short for $FRIENDLY_NAME) provides a C++ interface to LAPACK routines SGELSD, DGELSD, CGELSD, and ZGELSD. gelsd computes the minimum-norm solution to a real 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.

Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X.

The problem is solved in three steps: (1) Reduce the coefficient matrix A to bidiagonal form with Householder tranformations, reducing the original problem into a "bidiagonal least squares problem" (BLS) (2) Solve the BLS using a divide and conquer approach. (3) Apply back all the Householder tranformations to solve the original least squares problem.

The effective rank of A is determined by treating as zero those singular values which are less than RCOND times the largest singular value.

The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

The selection of the LAPACK routine is done during compile-time, and is determined by the type of values contained in type MatrixA. The type of values is obtained through the value_type meta-function typename value_type<MatrixA>::type. The dispatching table below illustrates to which specific routine the code path will be generated.

Definition

Defined in header boost/numeric/bindings/lapack/driver/gelsd.hpp.

Parameters or Requirements on Types

Complexity

Example

#include <boost/numeric/bindings/lapack/driver/gelsd.hpp>
using namespace boost::numeric::bindings;

lapack::gelsd( x, y, z );

this will output

[5] 0 1 2 3 4 5

Notes

See Also