If $A$ is a $m \times n$ matrix whose columns are linearly independent, the $QR$ decomposition of $A$ is $QR = A$, where $Q$ is an $m \times m$ orthogonal matrix and $R$ is an upper triangular $m \times n$ matrix.
i1 : A = random(RR^5, RR^3)
o1 = | .892712 .714827 .909047 |
| .673395 .89189 .314897 |
| .29398 .231053 .0741835 |
| .632944 .461944 .808694 |
| .0258884 .775187 .362835 |
5 3
o1 : Matrix RR <--- RR
53 53
|
i2 : (Q,R) = QRDecomposition A
o2 = (| -.677131 .143091 .247563 |, | -1.31837 -1.22811 -1.1883 |)
| -.510777 -.324257 -.666994 | | 0 -.816018 -.175579 |
| -.222987 .0524494 -.347041 | | 0 0 .523234 |
| -.480095 .156448 .507738 |
| -.0196366 -.92041 .339994 |
o2 : Sequence
|
$R$ is upper triangular, and $Q$ is (close to) orthogonal.
i3 : R
o3 = | -1.31837 -1.22811 -1.1883 |
| 0 -.816018 -.175579 |
| 0 0 .523234 |
3 3
o3 : Matrix RR <--- RR
53 53
|
i4 : (transpose Q) * Q
o4 = | 1 -4.16334e-17 -2.77556e-17 |
| -4.16334e-17 1 2.15106e-16 |
| -2.77556e-17 2.15106e-16 1 |
3 3
o4 : Matrix RR <--- RR
53 53
|
i5 : clean(1e-10, oo)
o5 = | 1 0 0 |
| 0 1 0 |
| 0 0 1 |
3 3
o5 : Matrix RR <--- RR
53 53
|
i6 : R - (transpose Q) * A
o6 = | 0 0 6.66134e-16 |
| 0 1.11022e-16 -1.11022e-16 |
| 1.11022e-16 1.94289e-16 -2.22045e-16 |
3 3
o6 : Matrix RR <--- RR
53 53
|
i7 : clean(1e-10, oo)
o7 = | 0 0 0 |
| 0 0 0 |
| 0 0 0 |
3 3
o7 : Matrix RR <--- RR
53 53
|
If the input is a MutableMatrix, then so are the output matrices.
i8 : A = mutableMatrix(RR_53, 13, 5); |
i9 : fillMatrix A
o9 = | .706096 .606588 .605659 .174853 .370833 |
| .127435 .848005 .96518 .626892 .339222 |
| .254482 .191734 .681683 .350611 .062212 |
| .741046 .403215 .914199 .379495 .465736 |
| .108386 .615911 .887381 .237252 .40273 |
| .348931 .0147867 .169813 .116721 .164647 |
| .562428 .223028 .965004 .444183 .713493 |
| .246268 .388829 .0647412 .644366 .909537 |
| .153346 .557119 .877846 .194945 .566034 |
| .830833 .873708 .0340514 .518585 .305423 |
| .538554 .7037 .507989 .987173 .732358 |
| .873665 .681869 .150294 .568273 .562839 |
| .415912 .276259 .656391 .184779 .629991 |
o9 : MutableMatrix
|
i10 : (Q,R) = QRDecomposition A
o10 = (| -.373219 .00053571 -.0145186 .434826 -.0273305 |, | -1.89191
| -.0673582 -.640249 -.160526 -.10301 .310349 | | 0
| -.134511 .0235031 -.312596 -.211448 .462328 | | 0
| -.391693 .202931 -.312943 .0456648 .146375 | | 0
| -.0572895 -.453214 -.233381 .214532 -.100994 | | 0
| -.184434 .24735 -.0733514 -.0385991 .0314994 |
| -.297281 .225982 -.452339 -.214843 -.131288 |
| -.130169 -.153513 .179847 -.466939 -.58274 |
| -.0810536 -.368715 -.2545 .236721 -.32996 |
| -.439151 -.138065 .495715 .204262 .184438 |
| -.284662 -.208589 .0850958 -.577067 .107535 |
| -.461791 .0602065 .325632 .0156234 -.0364137 |
| -.219838 .070582 -.240391 .103952 -.379431 |
-----------------------------------------------------------------------
-1.62694 -1.56271 -1.38193 -1.52288 |)
-1.15333 -.947084 -.695353 -.566482 |
0 -1.57949 -.0484065 -.388761 |
0 0 -.784772 -.515486 |
0 0 0 -.778323 |
o10 : Sequence
|
i11 : Q*R-A
o11 = | -2.22045e-16 0 5.55112e-16 2.22045e-16 4.44089e-16 |
| 0 1.11022e-16 -2.22045e-16 4.44089e-16 -1.66533e-16 |
| 0 2.77556e-17 -1.11022e-16 -2.22045e-16 -1.66533e-16 |
| 1.11022e-16 0 2.22045e-16 0 0 |
| 0 1.11022e-16 -1.11022e-16 2.77556e-17 -1.11022e-16 |
| 0 5.55112e-17 1.66533e-16 -4.16334e-17 0 |
| 0 5.55112e-17 2.22045e-16 -5.55112e-17 1.11022e-16 |
| 0 0 0 1.11022e-16 -1.11022e-16 |
| 0 0 0 1.11022e-16 0 |
| 0 1.11022e-16 1.11022e-16 1.11022e-16 1.66533e-16 |
| 0 0 0 -1.11022e-16 -1.11022e-16 |
| -1.11022e-16 0 2.22045e-16 1.11022e-16 3.33067e-16 |
| -5.55112e-17 5.55112e-17 2.22045e-16 -5.55112e-17 1.11022e-16 |
o11 : MutableMatrix
|
i12 : clean(1e-10,oo)
o12 = | 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
o12 : MutableMatrix
|
This function works by calling lapack routines, and so only uses the first 53 bits of precision. Lapack also has a way of returning an encoded pair of matrices that contain enough information to reconstruct $Q, R$.
If the matrices are over higher precision real or complex fields, such as $RR_100$, this extra precision is not used in the computation
The object QRDecomposition is a method function.