QR decomposition - MATLAB qr (2024)

Table of Contents

Syntax Description Examples Q-Less QR Decomposition Full QR Decomposition of Matrix Permuted QR Decomposition Solve Linear System with Economy-Size QR Factors Solve Sparse Linear System Solve Rectangular Sparse Linear System Input Arguments A — Input matrix matrix S — Input coefficient matrix matrix B — Right-hand side matrix matrix outputForm — Shape of permutation output "matrix" (default) | "vector" Output Arguments Q — Orthogonal factor matrix R — Upper-triangular factor matrix P — Permutation information matrix | vector C — Linear system factor matrix Tips References Extended Capabilities C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™. GPU Code Generation Generate CUDA® code for NVIDIA® GPUs using GPU Coder™. Thread-Based Environment Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool. GPU Arrays Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™. Distributed ArraysPartition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™. Version History R2022b: "0" syntax not recommended for economy-size decompositions R2022a: Option for economy-size decompositions R2022a: One-output syntax always returns upper-triangular factor See Also Topics MATLAB Command Americas Europe Asia Pacific FAQs References

QR decomposition

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Syntax

R = qr(A)

[Q,R] = qr(A)

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

[___] = qr(A,"econ")

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

[___] = qr(A,0)

[C,R] = qr(S,B)

[C,R,P] = qr(S,B)

[___] = qr(S,B,"econ")

[C,R,P] = qr(S,B,outputForm)

[___] = qr(S,B,0)

Description

example

R = qr(A) returns the upper-triangular R factor of the QR decomposition A = Q*R.

example

[Q,R] = qr(A) performs a QR decomposition on m-by-n matrix A such that A = Q*R. The factor R is an m-by-n upper-triangular matrix, and the factor Q is an m-by-m orthogonal matrix.

example

[Q,R,P] = qr(A) additionally returns a permutation matrix P such that A*P = Q*R. If A is full, the permutation matrix is chosen so that abs(diag(R)) is decreasing.

example

[___] = qr(A,"econ") produces an economy-size decomposition using any of the previous output argument combinations. The size of the outputs depends on the size of m-by-n matrix A:

If m > n, then qr computes only the first n columns of Q and the first n rows of R.
If m <= n, then the economy-size decomposition is the same as the regular decomposition.

example

[Q,R,P] = qr(A,outputForm) specifies whether to return the permutation information P as a matrix or a vector. For example, if outputForm is "vector", then A(:,P) = Q*R. The default value of outputForm is "matrix" such that A*P = Q*R.

[___] = qr(A,0) is equivalent to qr(A,"econ","vector"). The use of this syntax is not recommended. Use the "econ" option instead.

example

[C,R] = qr(S,B) computes C = Q'*B and the upper-triangular factor R. You can use C and R to compute a least-squares solution to the sparse linear system S*X = B with X = R\C.

example

[C,R,P] = qr(S,B) additionally returns a permutation matrix P that is chosen to reduce fill-in in R. You can use C, R, and P to compute a least-squares solution to the sparse linear system S*X = B with X = P*(R\C).

example

[___] = qr(S,B,"econ") produces an economy-size decomposition using any of the previous output argument combinations. The size of the outputs depends on the size of m-by-n sparse matrix S:

If m > n, then qr computes only the first n rows of C and R.
If m <= n, then the economy-size decomposition is the same as the regular decomposition.

example

[C,R,P] = qr(S,B,outputForm) specifies whether to return the permutation information P as a matrix or vector. For example, if outputForm is "vector", then the least-squares solution to S*X = B is X(P,:) = R\C. The default value of outputForm is "matrix" such that the least-squares solution to S*X = B is X = P*(R\C).

[___] = qr(S,B,0) is equivalent to qr(S,B,"econ","vector"). The use of this syntax is not recommended. Use the "econ" option instead.

Examples

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Q-Less QR Decomposition

Open Live Script

Find the QR decomposition of the 5-by-5 magic square matrix. Specify one output argument to return just the upper-triangular factor.

A = magic(5);R = qr(A)

R = 5×5 -32.4808 -26.6311 -21.3973 -23.7063 -25.8615 0 19.8943 12.3234 1.9439 4.0856 0 0 -24.3985 -11.6316 -3.7415 0 0 0 -20.0982 -9.9739 0 0 0 0 -16.0005

Full QR Decomposition of Matrix

Open Live Script

Compute the full QR decomposition of a magic square test matrix by specifying two output arguments.

A = magic(5);[Q,R] = qr(A)

Q = 5×5 -0.5234 0.5058 0.6735 -0.1215 -0.0441 -0.7081 -0.6966 -0.0177 0.0815 -0.0800 -0.1231 0.1367 -0.3558 -0.6307 -0.6646 -0.3079 0.1911 -0.4122 -0.4247 0.7200 -0.3387 0.4514 -0.4996 0.6328 -0.1774

R = 5×5 -32.4808 -26.6311 -21.3973 -23.7063 -25.8615 0 19.8943 12.3234 1.9439 4.0856 0 0 -24.3985 -11.6316 -3.7415 0 0 0 -20.0982 -9.9739 0 0 0 0 -16.0005

Verify that $A = QR$ , within machine precision.

norm(A-Q*R)

ans = 9.6765e-15

Permuted QR Decomposition

Open Live Script

Specify three output arguments to return a permutation matrix or vector that reduces fill-in in the R factor of the QR decomposition.

Compute the QR decomposition of the west0479 sparse matrix. Specify three outputs to return a permutation matrix that satisfies $AP = QR$ .

load west0479A = west0479;[Q,R,P] = qr(A);

Verify that A*P = Q*R for the permutation matrix P, within machine precision.

norm(A*P-Q*R,"fro")

ans = 3.5451e-10

Now specify the "vector" option to return p as a permutation vector.

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

Verify that A(:,p) = Q*R for the permutation vector p, within machine precision.

norm(A(:,p) - Q*R,"fro")

ans = 3.5451e-10

Verify that the use of a permutation matrix or permutation vector in the decomposition results in an R factor with fewer nonzeros for sparse inputs compared to a nonpermuted decomposition.

[Q1,R1] = qr(A);spy(R1)

spy(R)

The results show that the permuted decomposition produces an R factor with substantially fewer nonzeros.

Solve Linear System with Economy-Size QR Factors

Open Live Script

Use the economy-size QR decomposition of a coefficient matrix to solve the linear system $Ax = b$ .

Create a 10-by-5 coefficient matrix by using the first five columns of magic(10). For the right-hand side of the linear equation $Ax = b$ , use the row sums of the matrix. With this setup, the solution to the equation $x$ should be a vector of ones.

A = magic(10);A = A(:,1:5)

A = 10×5 92 99 1 8 15 98 80 7 14 16 4 81 88 20 22 85 87 19 21 3 86 93 25 2 9 17 24 76 83 90 23 5 82 89 91 79 6 13 95 97 10 12 94 96 78 11 18 100 77 84

b = sum(A,2)

b = 10×1 215 215 215 215 215 290 290 290 290 290

Compute the economy-size QR decomposition of A. Then solve the linear system $QRx = b$ with x(p,:) = R\(Q\b). Because Q is orthogonal, this equation is the same as x(p,:) = R\(Q'*b).

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

Q = 10×5 -0.0050 -0.4775 -0.0504 0.5193 0.0399 -0.0349 -0.5001 -0.0990 -0.1954 -0.2006 -0.4384 0.1059 -0.4660 0.4464 0.0628 -0.0947 -0.4151 -0.2923 -0.2542 0.5274 -0.1246 -0.4117 -0.2812 -0.1326 -0.4130 -0.3787 0.0209 0.2702 0.4697 0.0390 -0.4085 -0.0017 0.2217 -0.2450 -0.2015 -0.0648 -0.3925 0.6939 0.0669 0.1225 -0.4683 0.0833 0.0283 -0.3038 0.5265 -0.4982 0.0867 0.0394 -0.1822 -0.4138

R = 5×5 -200.7112 -55.5026 -167.6040 -84.7237 -168.7997 0 -192.1053 -40.3557 -152.4040 -39.2814 0 0 101.3180 -89.4254 96.0172 0 0 0 41.0248 -14.9083 0 0 0 0 24.6386

p = 1×5 3 1 5 2 4

x(p,:) = R\(Q\b)

x = 5×1 1.0000 1.0000 1.0000 1.0000 1.0000

Make a semilog plot of the diagonal of R to confirm that the permuted decomposition produces an R factor with abs(diag(R)) decreasing. Plot the singular values of A in the same plot for comparison. In practice, the diagonal values of R behave in a similar way to the singular values of A. Therefore, you can use the diagonal values of R as a measure for how close to singular the matrix A is.

semilogy(abs(diag(R)),"-o")hold onsemilogy(svd(A),"r-o")legend("Diagonal of R","Singular Values of A")

Solve Sparse Linear System

Open Live Script

Solve a sparse linear system and use the results to see how much of vector b lies in the column space of S.

Create a random 500-by-20 sparse matrix with 10% density and a vector of ones. Use qr to factorize the matrix into the factors R and C = Q'*b.

S = sprand(500,20,0.1);b = ones(500,1);[C,R] = qr(S,b,"econ");

Use the results to solve $Sx = b$ with x = R\C.

x = R\C;

Consider the identity ${‖ b ‖}^{2} = {‖ Sx - b ‖}^{2} + {‖ C ‖}^{2}$ .

Dividing through by the norm of b, you get a new identity that shows how much of b lies in the column space of S:

$\frac{{‖ Sx - b ‖}^{2}}{{‖ b ‖}^{2}} + \frac{{‖ C ‖}^{2}}{{‖ b ‖}^{2}} = 1$ .

The first term tells how much of b does not lie in the column space of S, while the second term tells how much of b does lie in the column space of S.

t1 = norm(S*x-b)^2/norm(b)^2

t1 = 0.4000

t2 = norm(C)^2/norm(b)^2

t2 = 0.6000

Solve Rectangular Sparse Linear System

Open Live Script

Use qr to solve the matrix equation $Sx = B$ with a rectangular sparse coefficient matrix $S$ .

Load the west0479 sparse matrix and use the first 200 columns as the rectangular coefficient matrix in a linear system. For the right-hand side of the equation, use the row sums of $S$ . With this setup, the solution to $Sx = B$ is a vector of ones.

load west0479S = west0479(:,1:200);B = sum(S,2);

Solve $Sx = B$ using qr with two inputs and three outputs. The solution to the linear system is x = P*(R\C).

[C,R,P] = qr(S,B);x = P*(R\C);

Verify that $Sx - B = 0$ , within machine precision.

norm(S*x-B)

ans = 8.3874e-11

Note: To calculate the upper-triangular factor R and permutation matrix P, but avoid computing the orthogonal matrix Q (which is often the most computationally expensive part of a call to qr), you can specify B as an empty matrix:

emptyB = zeros(size(S,1),0);[~,R,P] = qr(S,emptyB);

Input Arguments

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`A` — Input matrix
matrix

Input matrix, specified as a full or sparse matrix.

Data Types: single | double
Complex Number Support: Yes

`S` — Input coefficient matrix
matrix

Input coefficient matrix, specified as a sparse matrix. With two input matrices, qr computes a least-squares solution to the linear system S*X = B.

Data Types: double
Complex Number Support: Yes

`B` — Right-hand side matrix
matrix

Right-hand side matrix, specified as a full or sparse matrix. With two input matrices, qr computes C = Q'*B, which you can use to solve the linear system S*X = B.

Data Types: single | double
Complex Number Support: Yes

`outputForm` — Shape of permutation output
`"matrix"` (default) | `"vector"`

Shape of permutation output, specified as "matrix" or "vector". This flag controls whether the permutation output P is returned as a permutation matrix or permutation vector. You must specify three output arguments to qr to use this option.

If outputForm is "vector", then P is a permutation vector that satisfies A(:,P) = Q*R.
The default value of outputForm is "matrix" such that A*P = Q*R.

Example: [Q,R,P] = qr(A,"vector")

Output Arguments

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`Q` — Orthogonal factor
matrix

Orthogonal factor, returned as a matrix that satisfies A = Q*R for an m-by-n matrix A.

For full decompositions, qr(A) returns Q as an m-by-m orthogonal matrix satisfying $Q^{H} Q = Q Q^{H} = I_{m}$ .
For rectangular A with m > n, the economy-sized decomposition qr(A,"econ") computes only the first n columns of Q and first n rows of R. The columns of Q form an orthonormal basis for the column space of A.

Different machines and releases of MATLAB^® can produce different columns in Q that are still numerically accurate. Corresponding rows and columns in Q and R can flip their signs, since this does not affect the value of the expression A = Q*R.

`R` — Upper-triangular factor
matrix

Upper-triangular factor, returned as a matrix that satisfies A = Q*R. The diagonal of R is in decreasing order when A is full and three outputs are specified, [Q,R,P] = qr(A).

`P` — Permutation information
matrix | vector

Permutation information, returned as a matrix or vector. The shape of P depends on the value of outputForm. Also, qr selects P to satisfy different criteria depending on whether the first input matrix is full or sparse:

Full — qr selects P so that abs(diag(R)) is decreasing.
Sparse — qr selects P to reduce fill-in in R.

`C` — Linear system factor
matrix

Linear system factor, returned as a matrix that satisfies C = Q'*B. The least-squares solution to S*X = B is X = R\C. If the permutation output P is specified, then the solution is either X = P*(R\C) or X(P,:) = R\C, depending on the value of outputForm:

If outputForm is "vector", then the least-squares solution to S*X = B is X(P,:) = R\C.
The default value of outputForm is "matrix" such that the least-squares solution to S*X = B is X = P*(R\C).

Tips

To solve multiple linear systems involving the same coefficient matrix, use decomposition objects.
For the syntax [C,R] = qr(S,B), the value of X = R\C is a least-squares solution to S*X = B only when S does not have low rank.

References

[1] Anderson, E., ed. LAPACK Users’ Guide. 3rd ed. Software, Environments, Tools. Philadelphia: Society for Industrial and Applied Mathematics, 1999. https://doi.org/10.1137/1.9780898719604.

[2] Davis, Timothy A. “Algorithm 915, SuiteSparseQR: Multifrontal Multithreaded Rank-Revealing Sparse QR Factorization.” ACM Transactions on Mathematical Software 38, no. 1 (November 2011): 1–22. https://doi.org/10.1145/2049662.2049670.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

Code generation does not support sparse matrix inputs for this function.

GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Usage notes and limitations:

Code generation does not support sparse matrix inputs for this function.
Code generation might return a different QR factorization than MATLAB. You can verify the Q and R values by using the equation Q*R = A.

Thread-Based Environment
Run code in the background using MATLAB® `backgroundPool` or accelerate code with Parallel Computing Toolbox™ `ThreadPool`.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

Sparse inputs are not supported.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays
Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

Usage notes and limitations:

The input A must be nonsparse.

For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced before R2006a

expand all

R2022b: "0" syntax not recommended for economy-size decompositions

The syntaxes [___] = qr(A,0) and [___] = qr(S,B,0) will continue to be supported, but are no longer recommended. Use the "econ" option to perform economy-size decompositions instead.

R2022a: Option for economy-size decompositions

Use the "econ" option to calculate economy-size decompositions with qr. The functionality is the same as qr(A,0) unless a third output is specified.

R2022a: One-output syntax always returns upper-triangular factor

The syntax R = qr(A) always returns R as an upper-triangular matrix, regardless of whether A is full or sparse. Previously, for full A, the one-output syntax returned an R matrix with Householder vectors located in the lower-triangular portion of the matrix. These vectors partially defined the Q matrix.

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FAQs

QR decomposition - MATLAB qr? ›

The QR decomposition, also known as the QR factorization, expresses an m-by-n matrix A as A = Q*R . For the full decomposition, Q is an m-by-m unitary matrix, and R is an m-by-n upper triangular matrix. If the components of A are real numbers, then Q is an orthogonal matrix.

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What is decomposition in MATLAB? ›

dA = decomposition( A ) returns a decomposition of matrix A that you can use to solve linear systems more efficiently. The decomposition type is automatically chosen based on the properties of the input matrix. dA = decomposition( A , type ) specifies the type of decomposition to perform.

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How do you find the full QR decomposition? ›

There are really two kinds of QR decomposition: there is the full QR decomposition, where A is m×n, Q is m×m, and R is m×n. There is also the reduced QR decomposition, where A is m×n with rank r, Q is m×r, and R is r×n.

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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.

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What does QR stand for in QR decomposition? ›

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.

Read On ›

How does decomposition method work? ›

Decomposition is a general approach to solving a problem by breaking it up into smaller ones and solving each of the smaller ones separately, either in parallel or sequentially. (When it is done sequentially, the advantage comes from the fact that problem complexity grows more than linearly.)

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How does decomposition work? ›

Decomposition is a complex process. Organic matter is broken down into carbon dioxide and the mineral forms of nutrients like nitrogen. It is also converted into fungi and bacteria through these organisms feeding on the organic material and reproducing.

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What are the advantages of QR decomposition? ›

Advantages and Limitations

QR-decomposition is favored for its numerical stability and the orthogonality of matrix Q, which preserves the norm during transformations.

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Does QR decomposition always exist? ›

The QR decompostion always exists, even if the matrix does not have full rank, so the constructor will never fail. The primary use of the QR decomposition is in the least squares solution of nonsquare systems of simultaneous linear equations.

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Is the QR decomposition unique? ›

The QR decomposition is unique. satisfying the stated properties are unique. Thus, the two matrices involved in the QR decomposition are unique.

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How does Matlab compute QR? ›

[ Q , R ] = qr( A ) performs a QR decomposition on m -by- n matrix A such that A = Q*R . The factor R is an m -by- n upper-triangular matrix, and the factor Q is an m -by- m orthogonal matrix. [ Q , R , P ] = qr( A ) additionally returns a permutation matrix P such that A*P = Q*R .

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Who invented QR decomposition? ›

The QR algorithm was developed in the late 1950s by John G. F. Francis and by Vera N. Kublanovskaya, working independently. The basic idea is to perform a QR decomposition, writing the matrix as a product of an orthogonal matrix and an upper triangular matrix, multiply the factors in the reverse order, and iterate.

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How to use QR decomposition to find eigenvalues? ›

3.6. Using QR Decomposition to Compute Eigenvalues

Set A 0 = A and form. ...
Form A 1 = R 0 Q 0 . ...
Form A 1 = Q 1 R 1 (i.e., form the decomposition of ).
Form A 2 = R 1 Q 1 and then A 2 = Q 2 R 2 .
Iterate to convergence.
Compute eigenvalues of and compare them to the diagonal values of the limiting found from this process.

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What does decomposition mean in programming? ›

Decomposition is one of the four cornerstones of Computer Science. It involves breaking down a complex problem or system into smaller parts that are more manageable and easier to understand. The smaller parts can then be examined and solved, or designed individually, as they are simpler to work with.

What does it mean to decompose a function? ›

In engineering, functional decomposition is the process of resolving a functional relationship into its constituent parts in such a way that the original function can be reconstructed (i.e., recomposed) from those parts.

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What is decomposition in machine learning? ›

Matrix decomposition, also known as matrix factorization or matrix factor decomposition, is a fundamental concept in the field of machine learning. It involves breaking down a matrix into its constituent parts to gain a deeper understanding and extract meaningful information from the data it represents.

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What does decomposition mean in signal processing? ›

The goal of signal decomposition is extraction and separation of signal components from composite signals, which should preferably be related to semantic units. Examples for this are distinct objects in images or video, video shots, melody sequences in music, spoken words or sentences in speech signals.

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