gels(A, B[, trans=’N’])
Solves least-squares and least-norm problems with a full rank m by n matrix A.
If m is less than or equal to n, gels() solves the least-norm problem
If m is less than or equal to n, gels() solves the least-squares problem
If m is less than or equal to n, gels() solves the least-squares problem
A and B must have the same typecode (’d’ or ’z’). trans = ’T’ is not allowed if A is complex. On exit, the solution X is stored as the leading submatrix of B. The array A is overwritten with details of the QR or the LQ factorization of A. Note that gels() does not check whether A is full rank.
geqrf(A, tau)
QR factorization of a real or complex matrix A:
If A is m by n, then Q is m by m and orthogonal/unitary, and R is m by n and upper triangular (if m is greater than or equal to n), or upper trapezoidal (if m is less than or equal to n). tau is a matrix of the same type as A and of length at least min{m,n}. On exit, R is stored in the upper triangular part of A. The matrix Q is stored as a product of min{m,n} elementary reflectors in the first min{m,n} columns of A and in tau.
ormqr(A, tau, C[, side=’L’[, trans=’N’]])
Product with a real orthogonal matrix:
where Q is square and orthogonal. Q is stored in A and tau as a product of min{A.size[0], A.size[1]} elementary reflectors, as computed by geqrf().
unmqr(A, tau, C[, side=’L’[, trans=’N’]])
Product with a real orthogonal or complex unitary matrix:
Q is square and orthogonal or unitary. Q is stored in A and tau as a product of min{A.size[0], A.size[1]} elementary reflectors, as computed by geqrf(). The arrays A, tau and C must have the same type. trans = ’T’ is only allowed if the typecode is ’d’.
In the following example, we solve a least-squares problem by a direct call to gels(), and by separate calls to geqrf(), ormqr(), and trtrs().
>>> from cvxopt import random, blas, lapack
>>> from cvxopt.base import matrix >>> m, n = 10, 5 >>> A, b = random.normal(m,n), random.normal(m,1) >>> x1 = +b >>> lapack.gels(+A, x1) # x1[:n] minimizes ||A*x1[:n] - b||_2 >>> tau = matrix(0.0, (n,1)) >>> lapack.geqrf(A, tau) # A = [Q1, Q2] * [R1; 0] >>> x2 = +b >>> lapack.ormqr(A, tau, x2, trans=’T’) # x2 := [Q1, Q2]’ * b >>> lapack.trtrs(A[:n,:], x2, uplo=’U’) # x2[:n] := R1^{-1}*x2[:n] >>> blas.nrm2(x1[:n] - x2[:n]) 3.0050798580569307e-16 |