## QR Decomposition

A general rectangular M-by-N matrix A has a QR decomposition into the product of an orthogonal M-by-M square matrix Q (where Q^T Q = I) and an M-by-N right-triangular matrix R,

    A = Q R

This decomposition can be used to convert the linear system A x = b into the triangular system R x = Q^T b, which can be solved by back-substitution. Another use of the QR decomposition is to compute an orthonormal basis for a set of vectors. The first N columns of Q form an orthonormal basis for the range of A, ran(A), when A has full column rank.

Function: int gsl_linalg_QR_decomp (gsl_matrix * A, gsl_vector * tau)
  This function factorizes the M-by-N matrix A into the QR decomposition A = Q R. On output the diagonal and upper triangular part of the input matrix contain the matrix R. The vector tau and the columns of the lower triangular part of the matrix A contain the Householder coefficients and Householder vectors which encode the orthogonal matrix Q. The vector tau must be of length k=\min(M,N). The matrix Q is related to these components by, Q = Q_k ... Q_2 Q_1 where Q_i = I - \tau_i v_i v_i^T and v_i is the Householder vector v_i = (0,...,1,A(i+1,i),A(i+2,i),...,A(m,i)). This is the same storage scheme as used by LAPACK.

  The algorithm used to perform the decomposition is Householder QR (Golub & Van Loan, Matrix Computations, Algorithm 5.2.1).

Function: int gsl_linalg_QR_solve (const gsl_matrix * QR, const gsl_vector * tau, const gsl_vector * b, gsl_vector * x)
  This function solves the system A x = b using the QR decomposition of A into (QR, tau) given by `gsl_linalg_QR_decomp`.

Function: int gsl_linalg_QR_svx (const gsl_matrix * QR, const gsl_vector * tau, gsl_vector * x)
  This function solves the system A x = b in-place using the QR decomposition of A into (QR,tau) given by `gsl_linalg_QR_decomp`. On input x should contain the right-hand side b, which is replaced by the solution on output.

Function: int gsl_linalg_QR_lssolve (const gsl_matrix * QR, const gsl_vector * tau, const gsl_vector * b, gsl_vector * x, gsl_vector * residual)
  This function finds the least squares solution to the overdetermined system A x = b where the matrix A has more rows than columns. The least squares solution minimizes the Euclidean norm of the residual, ||Ax - b||.The routine uses the QR decomposition of A into (QR, tau) given by `gsl_linalg_QR_decomp`. The solution is returned in x. The residual is computed as a by-product and stored in residual.

Function: int gsl_linalg_QR_QTvec (const gsl_matrix * QR, const gsl_vector * tau, gsl_vector * v)
  This function applies the matrix Q^T encoded in the decomposition (QR,tau) to the vector v, storing the result Q^T v in v. The matrix multiplication is carried out directly using the encoding of the Householder vectors without needing to form the full matrix Q^T.

Function: int gsl_linalg_QR_Qvec (const gsl_matrix * QR, const gsl_vector * tau, gsl_vector * v)
  This function applies the matrix Q encoded in the decomposition (QR,tau) to the vector v, storing the result Q v in v. The matrix multiplication is carried out directly using the encoding of the Householder vectors without needing to form the full matrix Q.

Function: int gsl_linalg_QR_Rsolve (const gsl_matrix * QR, const gsl_vector * b, gsl_vector * x)
  This function solves the triangular system R x = b for x. It may be useful if the product b' = Q^T b has already been computed using `gsl_linalg_QR_QTvec`.

Function: int gsl_linalg_QR_Rsvx (const gsl_matrix * QR, gsl_vector * x)
  This function solves the triangular system R x = b for x in-place. On input x should contain the right-hand side b and is replaced by the solution on output. This function may be useful if the product b' = Q^T b has already been computed using `gsl_linalg_QR_QTvec`.

Function: int gsl_linalg_QR_unpack (const gsl_matrix * QR, const gsl_vector * tau, gsl_matrix * Q, gsl_matrix * R)
  This function unpacks the encoded QR decomposition (QR,tau) into the matrices Q and R, where Q is M-by-M and R is M-by-N.

Function: int gsl_linalg_QR_QRsolve (gsl_matrix * Q, gsl_matrix * R, const gsl_vector * b, gsl_vector * x)
  This function solves the system R x = Q^T b for x. It can be used when the QR decomposition of a matrix is available in unpacked form as (Q,R).

Function: int gsl_linalg_QR_update (gsl_matrix * Q, gsl_matrix * R, gsl_vector * w, const gsl_vector * v)
  This function performs a rank-1 update w v^T of the QR decomposition (Q, R). The update is given by Q'R' = Q R + w v^T where the output matrices Q' and R' are also orthogonal and right triangular. Note that w is destroyed by the update.

Function: int gsl_linalg_R_solve (const gsl_matrix * R, const gsl_vector * b, gsl_vector * x)
  This function solves the triangular system R x = b for the N-by-N matrix R.

Function: int gsl_linalg_R_svx (const gsl_matrix * R, gsl_vector * x)
  This function solves the triangular system R x = b in-place. On input x should contain the right-hand side b, which is replaced by the solution on output.