# QRDecomposition -- compute a QR decomposition of a real matrix

## Synopsis

• Usage:
(Q,R) = QRDecomposition A
• Inputs:
• A, , or , over $RR_{53}$
• Outputs:
• Q, ,
• R, , $(Q,R)$ is the QR-decomposition of $A$

## Description

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 $Q, R$.

## Caveat

If the matrices are over higher precision real or complex fields, such as $RR_100$, this extra precision is not used in the computation