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.