What is the QR decomposition system of linear equations?

In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthonormal matrix Q and an upper triangular matrix R .

Is QR decomposition is the same as LU decomposition?

Another decomposition method is QR decomposition. One advantage of QR decomposition over LU decomposition is that this method does not require that the decomposition be carried out on a square matrix .

When to use QR decomposition?

So-called QR-decompositions are useful for solving linear systems, eigenvalue problems and least squares approximations .

What is the formula for the QR factorization?

A = QR ⇒ QT A = QT QR = R. In other words, the formula R = QT A holds, no matter what m and n are. It doesn't matter if the matrix is square or not.

What is the QR decomposition of linear regression?

The QR decomposition is an approach of breaking a matrix down into its constituent elements . Where A is the matrix that we wish to decompose, Q a matrix with the size m x m, and R is an upper triangle matrix with the size m x n. The QR decomposition is a popular approach for solving the linear least squares equation.

What is the LU decomposition method applied to the linear system?

The LU decomposition was developed by Alan Turing as an alternative way carrying out Gaussian elimination through factorization of the coefficient matrix into a product of upper and lower triangular matrices , namely, A = LU [8] . The system is solved in two consecutive steps using the equations LY = B and UX = Y [9] .

What is the existence of QR decomposition?

A QR decomposition can be created for any matrix — it need not be square and it need not have full rank . The matrix Q is unitary, and R is upper triangular. Thus, each column of A can be expressed as a linear combination of the columns of Q, which form an orthonormal basis.

What is the solution QR to this system of linear equations 12q + 3r + 15?

Summary: The solution (q, r) to this system of linear equations 12q + 3r = 15 and - 4q - 4r = - 44 is (-2, 13) .

solving linear system with decomposition(A,'qr') and qr(A) produce ... (2024)

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Qiang Li on 24 Jul 2024 at 11:35

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Commented: Christine Tobler on 24 Jul 2024 at 16:41

Accepted Answer: Christine Tobler

post.mat

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Ran in:

load post.mat

x1 = decomposition(CA,'qr')\b_f;

Warning: Rank deficient, rank = 44, tol = 1.470347e-01.

[qq2,rr2,pp2] = qr(CA);

x2= pp2 * (rr2\(qq2'*b_f));

Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 3.100607e-25.

[qq3,rr3,pp3] = qr(CA,"econ","vector");

x3(pp3,:) = rr3\(qq3'*b_f);

Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 3.100607e-25.

any(x1-x2~=0)

ans = 1x3 logical array

1 1 1

any(x3-x2~=0)

ans = 1x3 logical array

0 0 0

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Accepted Answer

Christine Tobler on 24 Jul 2024 at 12:23

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Ran in:

post.mat

There are two reasons that the results don't match:

1) When the matrix is not full-rank, the QR-based solver in decomposition only uses the upper left triangle of the R matrix returned from QR. I have replicated this in output x3 in the code below.

2) The matrix Q in decomposition is stored in a more compact format (Householder reflectors) than the one returned by the QR function. This results in some differences on the level of round-off error.

load post.mat

x1 = decomposition(CA,'qr')\b_f;

Warning: Rank deficient, rank = 44, tol = 1.470347e-01.

[qq2,rr2,pp2] = qr(CA);

x2= pp2 * (rr2\(qq2'*b_f));

Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 3.100607e-25.

norm(x1 - x2)

ans = 4.3604e+09

rk = 44;

m = size(CA, 1);

nrhs = size(b_f, 2);

[qq2,rr2,pp2] = qr(CA);

x3 = pp2 * [(rr2(1:rk, 1:rk)\(qq2(:, 1:rk)'*b_f)); zeros(m-rk, nrhs)];

norm(x1 - x3)

ans = 1.2555e-13

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Qiang Li on 24 Jul 2024 at 14:05

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Ran in:

post.mat

Hi Christine,

I used the 'RankTolerance' option of decomposition, and compared the results. Is the 0.0024 difference coming from your second point? Thanks.

load post.mat

min(svd(CA))

ans = 3.2181e-12

x1 = decomposition(CA,'qr','RankTolerance',1e-12)\b_f;

rk = 60;

m = size(CA, 1);

nrhs = size(b_f, 2);

[qq2,rr2,pp2] = qr(CA);

x3 = pp2 * [(rr2(1:rk, 1:rk)\(qq2(:, 1:rk)'*b_f)); zeros(m-rk, nrhs)];

Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 3.100607e-25.

norm(x1 - x3)

ans = 0.0024

Christine Tobler on 24 Jul 2024 at 16:41

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Ran in:

post.mat

Yes, it's initially coming from point (2). This is amplified by the fact that matrix CA is ill-conditioned (max / min singular value is about 3e24), which means that the full matrix R is also ill-conditioned, and so the application of R\(Q'*b) takes a small difference in Q'*b and amplifies it a lot.

While RankTolerance=1e-12 doesn't seem particularly tiny, this is an absolute tolerance (not scaled by the maximum singular value of A), meaning it corresponds to about 1e-25 in relative terms.

load post.mat

max(svd(CA))

ans = 1.1161e+13

min(svd(CA))

ans = 3.2181e-12

cond(CA)

ans = 3.4681e+24

[qq2,rr2,pp2] = qr(CA);

cond(rr2)

ans = 3.4532e+24

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