GSL::Linalg::LU.decomp(A)
GSL::Matrix#LU_decomp
These method calculate the LU decomposition of the matrix. The returned
value is an array of [LU, perm, sign]
.
ex1)
include GSL::Linalg m = Matrix.new(...) lu, perm, sign = LU.decomp(m)
ex2)
lu, perm, sign = m.LU_decomp
GSL::Linalg::LU.solve(A, b)
GSL::Linalg::LU.solve(lu, perm, b)
GSL::Matrix#LU_solve(b)
GSL::Linalg::LU::LUMatrix#solve(perm, b)
The following is an example to solve a linear system
A x = b, b = [1, 2, 3, 4]
using LU decomposition.
ex1)
require 'gsl' A = Matrix.new([0.18, 0.60, 0.57, 0.96], [0.41, 0.24, 0.99, 0.58], [0.14, 0.30, 0.97, 0.66], [0.51, 0.13, 0.19, 0.85]) lu, perm, sign = A.LU_decomp p lu.class <- GSL::Linalg::LU::LUMatrix, subclass of the Matrix class b = [1, 2, 3, 4].to_gv <- "b" must be given as a GSL::Vector x = Linalg::LU.solve(lu, perm, b) p x <- The solution is returned by a vector.
ex2)
A = Matrix.new([0.18, 0.60, 0.57, 0.96], [0.41, 0.24, 0.99, 0.58], [0.14, 0.30, 0.97, 0.66], [0.51, 0.13, 0.19, 0.85]) p A.class <- GSL::Matrix x = Linalg::LU.solve(A, b) <- LU decomposition is calculated internally (A is not modified)
GSL::Linalg::LU.svx(A, b)
GSL::Linalg::LU.svx(lu, perm, b)
GSL::Matrix#svx(b)
GSL::Linalg::LU::LUMatrix#svx(perm, b)
GSL::Linalg::LU.refine(A, lu, perm, b, x)
GSL::Linalg::LU.invert(A)
GSL::Linalg::LU.invert(lu, perm)
GSL::Matrix#invert
GSL::Linalg::LU::LUMatrix#invert(perm)
GSL::Linalg::LU.det(A)
GSL::Linalg::LU.det(lu, signum)
GSL::Matrix#det
GSL::Linalg::LU::LUMatrix#det(signum)
GSL::Linalg::QR.decomp(A)
GSL::Matrix#QR_decomp
These compute QR decomposition of the matrix and return an array [QR, tau]. ex)
include GSL::Linalg m = Matrix.new(...) qr, tau = m.QR_decomp p qr.class <----- GSL::Linalg::QR::QRMatrix, subclass of GSL::Matrix p tau.class <----- GSK::Linalg::QR::TauVector, subclass of GSL::Vector
or
qr, tau = QR.decomp(m)
GSL::Linalg::QR.solve(A, b)
GSL::Linalg::QR.solve(QR, tau, b)
GSL::Matrix#QR_solve(b)
GSL::Linalg::QR::QRMatrix#solve(tau, b)
Solve the system A x = b using the QR decomposition.
ex1)
include GSL::Linalg m = Matrix.new(...) b = Vector.new(...) x = QR.solve(m, b)
ex2)
qr, tau = QR.decomp(m) x = QR.solve(qr, tau, b)
ex3)
x = m.QR_solve(b)
ex4)
x = qr.solve(tau, b)
GSL::Linalg::QR.svx(A, x)
GSL::Linalg::QR.svx(QR, tau, x)
GSL::Matrix#QR_svx(x)
GSL::Linalg::QR::QRMatrix#svx(tau, x)
GSL::Linalg::QR.unpack(QR, tau)
GSL::Linalg::QR::QRMatrix#unpack(tau)
[Q, R]
.GSL::Linalg::QR.QRsolve(Q, R, tau)
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.GSL::Linalg::QRPT.decomp(A)
GSL::Matrix#QRPT_decomp
These methods factorize the M-by-N matrix A into the QRP^T decomposition A = Q R P^T, and return an array [QR, tau, perm, signum]
.
ex1)
require("gsl") include GSL::Linalg m = Matrix.new(...) qr, tau, perm = QRPT.decomp(m) p qr.class <----- GSL::Linalg::QRPT::QRMatrix, subclass of GSL::Matrix
ex2)
qr, tau, perm = m.QROT_decomp
GSL::Linalg::QRPT.decomp2(A)
GSL::Matrix#QRPT_decomp2
These return an array [Q, R, tau, perm, signum]
.
ex)
q, r, tau, perm = QRPT.decomp2(m) p q.class <----- GSL::Linalg::QRPT::QMatrix p r.class <----- GSL::Linalg::QRPT::RMatrix
GSL::Linalg::QRPT.solve(m, b)
GSL::Linalg::QRPT.solve(qr, tau, perm, b)
GSL::Matrix#QRPT_solve(A, b)
GSL::Linalg::QRPT::QRMatrix#solve(qr, tau, perm, b)
These methods solve the system A x = b
using the QRP^T decomposition of
A into QR, tau, perm. The solution x is returned as a Vector.
ex1)
include GSL::Linalg m = Matrix.new(...) qr, tau, perm = m.QRPT_decomp b = Vector.new([1, 2, 3, 4]) x = QRPT.solve(qr, tau, perm, b)
ex2)
x = QRPT.solve(m, b)
ex3)
x = qr.solve(tau, p, b)
ex4)
x = m.QRPT_solve(b)
GSL::Linalg::QRPT.svx(m, b)
GSL::Linalg::QRPT.svx(qr, tau, perm, b)
GSL::Matrix#QRPT_svx(A, b)
A x = b
using the QRP^T decomposition of
A into QR, tau, perm. The input b is overwritten by the solution
x.GSL::Linalg::QRPT.QRsolve(q, r, tau, perm, b)
This method solves the system R P^T 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 obtained by the method decomp2
.
ex)
q, r, tau, perm = QRPT_decomp2 x = QRPT.QRsolve(q, r, perm, b)
GSL::Linalg::QRPT.update(q, r, perm, u, v)
GSL::Linalg::QRPT.Rsolve(qr, perm, b)
GSL::Linalg::QRPT::QRMatrix#Rsolve(perm, b)
GSL::Linalg::QRPT.Rsvx(qr, perm, b)
GSL::Linalg::QRPT::QRMatrix#Rsvx(perm, b)
GSL::Linalg::SV.decomp(A)
GSL::Matrix#SV_decomp
These methods factorize the M-by-N matrix A into the singular value
decomposition A = U S V^T
using the Golub-Reinsch SVD algorithm,
and return an array [U, V, S]
.
ex1)
include GSL::Linalg m = Matrix.new(...) u, v, s = SV.decomp(m) p u.class <----- GSL::Linalg::SV::UMatrix p v.class <----- GSL::Linalg::SV::VMatrix p s.class <----- GSL::Linalg::SV::SVector
ex2)
u, v, s = m.SV_decomp
GSL::Linalg::SV.decomp_mod(A)
GSL::Matrix#SV_decomp_mod
GSL::Linalg::SV.decomp_jacobi(A)
GSL::Matrix#SV_decomp_jacobi
GSL::Linalg::SV.solve(A, b)
GSL::Linalg::SV.solve(U, V, S, b)
GSL::Matrix#SV_solve(b)
These methods solve the system A x = b
using the singular value
decomposition U, S, V of A.
ex1)
include GSL::Linalg m = Matrix.new(...) b = Vector.new(...) u, v, s = SV.decomp(m) x = SV.solve(u, b, s, b)
ex2)
x = SV.solve(m, b)
ex3)
x = m.SV_solve(b)
GSL::Linalg::HH.solve(A, b)
GSL::Matrix#HH_solve(b)
A x = b
directly using Householder
transformations. GSL::Linalg::HH.svx(A, b)
GSL::Matrix#HH_svx(b)
A x = b
in-place directly using Householder
transformations. The input vector b is replaced by the solution.