QR decomposition of symbolic matrix (2024)

Compute the upper triangular R factor of the QR decomposition of the 4-by-4 Wilkinson's eigenvalue test matrix.

Create the 4-by-4 Wilkinson's eigenvalue test matrix.

A = sym(wilkinson(4))

A =
$$\left(\begin{array}{cccc}\frac{3}{2}& 1& 0& 0\\ 1& \frac{1}{2}& 1& 0\\ 0& 1& \frac{1}{2}& 1\\ 0& 0& 1& \frac{3}{2}\end{array}\right)$$

Return the R factor of the QR decomposition without returning the Q factor.

R = qr(A)

R =
$$\left(\begin{array}{cccc}\frac{\sqrt{13}}{2}& \frac{4\hspace{0.17em}\sqrt{13}}{13}& \frac{2\hspace{0.17em}\sqrt{13}}{13}& 0\\ 0& \frac{\sqrt{13}\hspace{0.17em}\sqrt{53}}{26}& \frac{10\hspace{0.17em}\sqrt{13}\hspace{0.17em}\sqrt{53}}{689}& \frac{2\hspace{0.17em}\sqrt{13}\hspace{0.17em}\sqrt{53}}{53}\\ 0& 0& \frac{\sqrt{53}\hspace{0.17em}\sqrt{381}}{106}& \frac{172\hspace{0.17em}\sqrt{53}\hspace{0.17em}\sqrt{381}}{20193}\\ 0& 0& 0& \frac{35\hspace{0.17em}\sqrt{381}}{762}\end{array}\right)$$

Compute the QR decomposition of the 3-by-3 Pascal matrix.

Create the 3-by-3 Pascal matrix.

A = sym(pascal(3))

A =
$$\left(\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 3& 6\end{array}\right)$$

Find the Q and R matrices representing the QR decomposition of A.

[Q,R] = qr(A)

Q =
$$\left(\begin{array}{ccc}\frac{\sqrt{3}}{3}& -\frac{\sqrt{2}}{2}& \frac{\sqrt{6}}{6}\\ \frac{\sqrt{3}}{3}& 0& -\frac{\sqrt{6}}{3}\\ \frac{\sqrt{3}}{3}& \frac{\sqrt{2}}{2}& \frac{\sqrt{6}}{6}\end{array}\right)$$

R =
$$\left(\begin{array}{ccc}\sqrt{3}& 2\hspace{0.17em}\sqrt{3}& \frac{10\hspace{0.17em}\sqrt{3}}{3}\\ 0& \sqrt{2}& \frac{5\hspace{0.17em}\sqrt{2}}{2}\\ 0& 0& \frac{\sqrt{6}}{6}\end{array}\right)$$

Verify that A = Q*R by using isAlways.

TF = isAlways(A == Q*R)

TF = 3x3 logical array 1 1 1 1 1 1 1 1 1

Use permutations to improve the numerical stability of the QR decomposition for floating-point matrices. The qr function returns permutation information either as a matrix or as a vector.

Set the number of significant decimal digits, used for variable-precision arithmetic, to 10. Approximate the 3-by-3 symbolic Hilbert matrix with floating-point numbers.

previoussetting = digits(10);A = vpa(hilb(3))

A =
$$\left(\begin{array}{ccc}1.0& 0.5& 0.3333333333\\ 0.5& 0.3333333333& 0.25\\ 0.3333333333& 0.25& 0.2\end{array}\right)$$

First, compute the QR decomposition of A without permutations.

[Q,R] = qr(A)

Q =
$$\left(\begin{array}{ccc}0.8571428571& -0.5016049166& 0.1170411472\\ 0.4285714286& 0.5684855721& -0.7022468832\\ 0.2857142857& 0.6520863915& 0.7022468832\end{array}\right)$$

R =
$$\left(\begin{array}{ccc}1.166666667& 0.6428571429& 0.45\\ 0& 0.1017143303& 0.1053370325\\ 0& 0& 0.003901371573\end{array}\right)$$

Compute the difference between A and Q*R. The computed Q and R matrices do not strictly satisfy the equality A = Q*R because of round-off errors.

E = A - Q*R

E =
$$\left(\begin{array}{ccc}-3.469446952e-18& -6.938893904e-18& -4.33680869e-18\\ 0& -1.734723476e-18& -8.67361738e-19\\ 0& -8.67361738e-19& -4.33680869e-19\end{array}\right)$$

To increase numerical stability of the QR decomposition, use permutations by specifying the syntax with three output arguments. For matrices that do not contain symbolic variables, expressions, or functions, this syntax triggers pivoting so that abs(diag(R)) in the returned matrix R is decreasing.

[Q,R,P] = qr(A)

Q =
$$\left(\begin{array}{ccc}0.8571428571& -0.4969293466& -0.1355261854\\ 0.4285714286& 0.5421047417& 0.7228063223\\ 0.2857142857& 0.6776309272& -0.6776309272\end{array}\right)$$

R =
$$\left(\begin{array}{ccc}1.166666667& 0.45& 0.6428571429\\ 0& 0.1054092553& 0.1016446391\\ 0& 0& 0.003764616262\end{array}\right)$$

P = 3×3 1 0 0 0 0 1 0 1 0

Check the equality A*P = Q*R. QR decomposition with permutations results in smaller round-off errors.

E = A*P - Q*R

E =
$$\left(\begin{array}{ccc}-3.469446952e-18& -4.33680869e-18& -6.938893904e-18\\ 0& -8.67361738e-19& -1.734723476e-18\\ 0& -4.33680869e-19& -1.734723476e-18\end{array}\right)$$

Now, return the permutation information as a vector by using the "vector" argument.

[Q,R,p] = qr(A,"vector")

Q =
$$\left(\begin{array}{ccc}0.8571428571& -0.4969293466& -0.1355261854\\ 0.4285714286& 0.5421047417& 0.7228063223\\ 0.2857142857& 0.6776309272& -0.6776309272\end{array}\right)$$

R =
$$\left(\begin{array}{ccc}1.166666667& 0.45& 0.6428571429\\ 0& 0.1054092553& 0.1016446391\\ 0& 0& 0.003764616262\end{array}\right)$$

p = 1×3 1 3 2

Verify that A(:,p) = Q*R.

E = A(:,p) - Q*R

E =
$$\left(\begin{array}{ccc}-3.469446952e-18& -4.33680869e-18& -6.938893904e-18\\ 0& -8.67361738e-19& -1.734723476e-18\\ 0& -4.33680869e-19& -1.734723476e-18\end{array}\right)$$

Exact symbolic computations do not result in round-off errors.

A = sym(hilb(3));[Q,R] = qr(A);E = A - Q*R

E =
$$\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$$

Restore the number of significant decimal digits to its previous setting.

digits(previoussetting)

You can solve systems of equations in matrix form by using qr.

Solve the system of equations A*X = b, where A is a 5-by-5 symbolic matrix and b is a 5-by-1 symbolic vector.

A = sym(invhilb(5))

A =
$$\left(\begin{array}{ccccc}25& -300& 1050& -1400& 630\\ -300& 4800& -18900& 26880& -12600\\ 1050& -18900& 79380& -117600& 56700\\ -1400& 26880& -117600& 179200& -88200\\ 630& -12600& 56700& -88200& 44100\end{array}\right)$$

b = sym([1:5]')

b =
$$\left(\begin{array}{c}1\\ 2\\ 3\\ 4\\ 5\end{array}\right)$$

Find matrices C and R such that C = Q'*b and A = Q*R.

[C,R] = qr(A,b);

Compute the solution X.

X = R\C

X =
$$\left(\begin{array}{c}5\\ \frac{71}{20}\\ \frac{197}{70}\\ \frac{657}{280}\\ \frac{1271}{630}\end{array}\right)$$

Verify that X is the solution of the system A*X = b by using isAlways.

tf = isAlways(A*X == b)

tf = 5x1 logical array 1 1 1 1 1

When solving systems of equations that contain floating-point numbers, use QR decomposition with a permutation matrix or vector.

Solve the system of equations A*X = b, where A is a 3-by-3 symbolic matrix with variable precision and b is a 3-by-1 symbolic vector with variable precision. Find matrices C and R such that C = Q'*b and A = Q*R.

previoussetting = digits(10);A = vpa([2 -3 -1; 1 1 -1; 0 1 -1]);b = vpa([2; 0; -1]);[C,R,P] = qr(A,b)

C =
$$\left(\begin{array}{c}-2.110579412\\ -0.2132007164\\ 0.7071067812\end{array}\right)$$

R =
$$\left(\begin{array}{ccc}3.31662479& 0.3015113446& -1.507556723\\ 0& 1.705605731& -1.492405014\\ 0& 0& 0.7071067812\end{array}\right)$$

P = 3×3 0 0 1 1 0 0 0 1 0

Compute the solution X.

X = P*(R\C)

X =
$$\left(\begin{array}{c}1.0\\ -0.25\\ 0.75\end{array}\right)$$

Alternatively, return the permutation information as a vector.

[C,R,p] = qr(A,b,"vector")

C =
$$\left(\begin{array}{c}-2.110579412\\ -0.2132007164\\ 0.7071067812\end{array}\right)$$

R =
$$\left(\begin{array}{ccc}3.31662479& 0.3015113446& -1.507556723\\ 0& 1.705605731& -1.492405014\\ 0& 0& 0.7071067812\end{array}\right)$$

p = 1×3 2 3 1

Compute the solution X with permutation vector p.

X(p,:) = R\C

X =
$$\left(\begin{array}{c}1.0\\ -0.25\\ 0.75\end{array}\right)$$

Restore the number of significant decimal digits to its previous setting.

digits(previoussetting)

Compute the economy-size QR decomposition by using the "econ" option.

Create a matrix that consists of the first two columns of the 4-by-4 Pascal matrix.

A = sym(pascal(4));A = A(:,1:2)

A =
$$\left(\begin{array}{cc}1& 1\\ 1& 2\\ 1& 3\\ 1& 4\end{array}\right)$$

Compute the QR decomposition for this matrix.

[Q,R] = qr(A)

Q =
$$\begin{array}{l}\left(\begin{array}{cccc}\frac{1}{2}& -{\sigma }_1& \frac{\sqrt{3}\hspace{0.17em}\sqrt{10}}{10}& 0\\ \frac{1}{2}& -\frac{\sqrt{5}}{10}& -\frac{2\hspace{0.17em}\sqrt{3}\hspace{0.17em}\sqrt{10}}{15}& \frac{\sqrt{6}}{6}\\ \frac{1}{2}& \frac{\sqrt{5}}{10}& -\frac{\sqrt{3}\hspace{0.17em}\sqrt{10}}{30}& -\frac{\sqrt{6}}{3}\\ \frac{1}{2}& {\sigma }_1& \frac{\sqrt{3}\hspace{0.17em}\sqrt{10}}{15}& \frac{\sqrt{6}}{6}\end{array}\right)\\ \\ \mathrm{where}\\ \\ \mathrm{}{\sigma }_1=\frac{3\hspace{0.17em}\sqrt{5}}{10}\end{array}$$

R =
$$\left(\begin{array}{cc}2& 5\\ 0& \sqrt{5}\\ 0& 0\\ 0& 0\end{array}\right)$$

Now, compute the economy-size QR decomposition for this matrix. Because the number of rows is greater than the number of columns in A, qr computes only the first 2 columns of Q and the first 2 rows of R.

[Q,R] = qr(A,"econ")

Q =
$$\left(\begin{array}{cc}\frac{1}{2}& -\frac{3\hspace{0.17em}\sqrt{5}}{10}\\ \frac{1}{2}& -\frac{\sqrt{5}}{10}\\ \frac{1}{2}& \frac{\sqrt{5}}{10}\\ \frac{1}{2}& \frac{3\hspace{0.17em}\sqrt{5}}{10}\end{array}\right)$$

R =
$$\left(\begin{array}{cc}2& 5\\ 0& \sqrt{5}\end{array}\right)$$

You can avoid complex conjugates in the result of QR decomposition by using the "real" option.

Create a matrix that includes a symbolic variable as an element.

syms xA = [1 2; 3 x]

A =
$$\left(\begin{array}{cc}1& 2\\ 3& x\end{array}\right)$$

Compute the QR decomposition of this matrix. By default, qr assumes that x represents a complex number, and therefore, the result contains expressions with the abs function.

[Q,R] = qr(A)

Q =
$$\begin{array}{l}\left(\begin{array}{cc}\frac{\sqrt{10}}{10}& -\frac{\frac{3\hspace{0.17em}x}{10}-\frac{9}{5}}{{\sigma }_1}\\ \frac{3\hspace{0.17em}\sqrt{10}}{10}& \frac{\frac{x}{10}-\frac{3}{5}}{{\sigma }_1}\end{array}\right)\\ \\ \mathrm{where}\\ \\ \mathrm{}{\sigma }_1=\sqrt{{\left|\frac{x}{10}-\frac{3}{5}\right|}^2+{\left|\frac{3\hspace{0.17em}x}{10}-\frac{9}{5}\right|}^2}\end{array}$$

R =
$$\left(\begin{array}{cc}\sqrt{10}& \frac{\sqrt{10}\hspace{0.17em}\left(3\hspace{0.17em}x+2\right)}{10}\\ 0& \sqrt{{\left|\frac{x}{10}-\frac{3}{5}\right|}^2+{\left|\frac{3\hspace{0.17em}x}{10}-\frac{9}{5}\right|}^2}\end{array}\right)$$

When you use the "real" option, qr assumes that all symbolic variables represent real numbers and can return shorter results.

[Q,R] = qr(A,"real")

Q =
$$\left(\begin{array}{cc}\frac{\sqrt{10}}{10}& -\frac{\frac{3\hspace{0.17em}x}{10}-\frac{9}{5}}{\sqrt{\frac{x^2}{10}-\frac{6\hspace{0.17em}x}{5}+\frac{18}{5}}}\\ \frac{3\hspace{0.17em}\sqrt{10}}{10}& \frac{\frac{x}{10}-\frac{3}{5}}{\sqrt{\frac{x^2}{10}-\frac{6\hspace{0.17em}x}{5}+\frac{18}{5}}}\end{array}\right)$$

R =
$$\left(\begin{array}{cc}\sqrt{10}& \frac{\sqrt{10}\hspace{0.17em}\left(3\hspace{0.17em}x+2\right)}{10}\\ 0& \sqrt{\frac{x^2}{10}-\frac{6\hspace{0.17em}x}{5}+\frac{18}{5}}\end{array}\right)$$

R — R factor of QR decomposition
upper triangular symbolic matrix

R factor of the QR decomposition, returned as an upper triangular symbolic matrix that satisfies A = Q*R for an m-by-n matrix A.

Q — Q factor of QR decomposition
unitary symbolic matrix

P — Permutation information
matrix | vector

Permutation information, returned as a matrix or vector of double-precision values. The shape of P depends on the value of outputForm.

C — Linear system factor
symbolic matrix

Linear system factor, returned as a symbolic matrix representing the solution of the matrix equation A*X = B, such that C = Q'*B and X = R\C. If you specify the permutation output P, then the solution is either X = P*(R\C) when outputForm is "matrix" or X(P,:) = R\C when outputForm is "vector".