the matrix to decompose into q and r
Perform backward substitution to solve for x in r*x = b.
Perform backward substitution to solve for x in r*x = b.
the upper triangular matrix
the constant vector
Show the flaw by printing the error message.
Show the flaw by printing the error message.
the method where the error occurred
the error message
Get the orthogonal q matrix.
Get bot the orthogonal q matrix and the right upper triangular r matrix.
Get the right upper triangular r matrix.
Compute the nullspace of matrix a: { x | a*x = 0 } using QR Decomposition qr*x = 0.
Compute the nullspace of matrix a: { x | a*x = 0 } using QR Decomposition qr*x = 0. Gives a basis of dimension n - rank for the nullspace
the rank of the matrix (number of linearly independent row vectors) FIX: should work, but it does not
Compute the nullspace of matrix a: { x | a*x = 0 } using QR Decomposition qr*x = 0.
Compute the nullspace of matrix a: { x | a*x = 0 } using QR Decomposition qr*x = 0. Gives only one vector in the nullspace.
Solve for x in a*x = b using QR Decomposition a = qr via r*x = q.
Solve for x in a*x = b using QR Decomposition a = qr via r*x = q.t * b.
the constant vector
This class is used to decompose an m by n matrix 'a' into an orthogonal m by n matrix 'q' and an n by n right upper triangular matrix 'r' such that a = q * r. It uses Gram-Schmidt orthogonalization. Note, orthogonal means that * q.t * q = I.
http://en.wikipedia.org/wiki/Gram–Schmidt_process (stabilized Gram–Schmidt orthonormalization)
,http://www.stat.wisc.edu/~larget/math496/qr.html