Queue Our Decomposition

Question

In this challenge I will ask you to find a QR decomposition of a square matrix. The QR decomposition of matrix A is two Matrices Q and R such that A = QR. In particular we are looking for Q to be an orthogonal matrix (that is Q^TQ=QQ^T=I where I is the multiplicative identity and ^T is the transpose) and R to be a upper triangular matrix (every value below its diagonal must be zero).

You will write code that takes a square matrix by any reasonable method and outputs a QR decomposition by any method. Many matrices have multiple QR decompositions however you only ever need output one.

Elements of your resultant matrices should be within two decimal places of an actual answer for every entry in the matrix.

This is a code-golf competition so answers will be scored in bytes with fewer bytes being a better score.

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Test Cases

These are only possible outputs, your outputs need not match all of these as long as they are valid.

0 0 0     1 0 0   0 0 0
0 0 0 ->  0 1 0   0 0 0
0 0 0     0 0 1 , 0 0 0

1 0 0     1 0 0   1 0 0
0 1 0 ->  0 1 0   0 1 0
0 0 1     0 0 1 , 0 0 1

1 2 3     1 0 0   1 2 3
0 3 1 ->  0 1 0   0 3 1
0 0 8     0 0 1 , 0 0 8

0 0 1     0 0 1   1 1 1
0 1 0 ->  0 1 0   0 1 0
1 1 1     1 0 0 , 0 0 1

0 0 0 0 1     0 0 0 0 1   1 0 0 0 1
0 0 0 1 0     0 0 0 1 0   0 1 1 1 0
0 0 1 0 0 ->  0 0 1 0 0   0 0 1 0 0
0 1 1 1 0     0 1 0 0 0   0 0 0 1 0
1 0 0 0 1     1 0 0 0 0 , 0 0 0 0 1

Answer 1

Julia, 2 bytes

qr

The function qr accepts a square matrix and returns a Tuple of matrices: Q and R.

Try it online!

Answer 2

Octave, 19 bytes

@(x)[[q,r]=qr(x),r]

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My first Octave answer \o/

Octave’s qr has quite a few alternatives in other languages that return both Q and R: QRDecomposition (Mathematica), matqr (PARI/GP), 128!:0 - if I recall correctly - (J), qr (R)...

Answer 3

Wolfram Language (Mathematica), 15 bytes

QRDecomposition

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I mean... what can I say?

Answer 4

R, 38 37 bytes

pryr::f(list(qr.R(q<-qr(m)),qr.Q(q)))

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Answer 5

SageMath, 27 bytes

lambda x:matrix(RDF,x).QR()

Answer 6

Python 2, 329 324 bytes

import fractions
I=lambda v,w:sum(a*b for a,b in zip(v,w))
def f(A):
 A,U=[map(fractions.Fraction,x)for x in zip(*A)],[]
 for a in A:
    u=a
    for v in U:u=[x-y*I(v,a)/I(v,v)for x,y in zip(u,v)]
    U.append(u)
 Q=[[a/I(u,u)**.5 for a in u]for u in U];return zip(*Q),[[I(e,a)*(i>=j)for i,a in enumerate(A)]for j,e in enumerate(Q)]

We must use fractions to ensure correct output, see https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process#Numerical_stability

Indentation used:

1. 1 space
2. 1 tab

Answer 7

Python with numpy, 28 bytes

import numpy
numpy.linalg.qr