Givens rotation

In numerical linear algebra, a Givens rotation is a rotation in the plane spanned by two coordinates axes. Givens rotations are named after Wallace Givens, who introduced them to numerical analysts in the 1950s while he was working at Argonne National Laboratory.

As action on matrices

A Givens rotation acting on a matrix from the left is a row operation, moving data between rows but always within the same column. Unlike the elementary operation of row-addition, a Givens rotation changes both of the rows addressed by it. To understand how it is a rotation, one may denote the elements of one target row by ${\displaystyle x_{1}}$ through ${\displaystyle x_{n}}$ and the elements of the other target row by ${\displaystyle y_{1}}$ through ${\displaystyle y_{n}}$: $${\displaystyle {\begin{bmatrix}\vdots &\vdots &\ddots &\vdots \\x_{1}&x_{2}&\dots &x_{n}\\\vdots &\vdots &\ddots &\vdots \\y_{1}&y_{2}&\dots &y_{n}\\\vdots &\vdots &\ddots &\vdots \end{bmatrix}}}$$ Then the effect of a Givens rotation is to rotate each subvector ${\displaystyle (x_{k},y_{k})}$ by the same angle. As with row-addition, algorithms often choose this angle so that one specific element becomes zero, and whatever happens in remaining columns is considered acceptable side-effects.

A Givens rotation acting on a matrix from the right is instead a column operation, moving data between two columns but always within the same row. As with action from the left, it rotates each subvector ${\displaystyle (x_{k},y_{k})}$ by the same angle, but here these named elements occur in the matrix as $${\displaystyle {\begin{bmatrix}\dots &x_{1}&\dots &y_{1}&\dots \\\dots &x_{2}&\dots &y_{2}&\dots \\\ddots &\vdots &\ddots &\vdots &\ddots \\\dots &x_{n}&\dots &y_{n}&\dots \end{bmatrix}}}$$ Some algorithms, especially those concerned with preserving matrix similarity, apply Givens rotations as a conjugate action: both rotating by one angle between two rows, and rotating by the same angle between the corresponding columns. In this case the effect on the four elements affected by both rotations is more complicated; a Jacobi rotation is such a conjugate action to the end of zeroing the two off-diagonal elements among these four.

The main use of Givens rotations in numerical linear algebra is to transform vectors or matrices into a special form with zeros in certain coefficients. This effect can, for example, be employed for computing the QR decomposition of a matrix. One advantage over Householder transformations is that they can easily be parallelised, and another is that often for very sparse matrices they have a lower operation count.

Matrix representation

A Givens rotation is represented by a matrix of the form

${\displaystyle G(i,j,\theta )={\begin{bmatrix}1&\cdots &0&\cdots &0&\cdots &0\\\vdots &\ddots &\vdots &&\vdots &&\vdots \\0&\cdots &c&\cdots &-s&\cdots &0\\\vdots &&\vdots &\ddots &\vdots &&\vdots \\0&\cdots &s&\cdots &c&\cdots &0\\\vdots &&\vdots &&\vdots &\ddots &\vdots \\0&\cdots &0&\cdots &0&\cdots &1\end{bmatrix}},}$

where c = cos θ and s = sin θ appear at the intersections ith and jth rows and columns. That is, for fixed i > j, the non-zero elements of Givens matrix are given by:

${\displaystyle {\begin{aligned}g_{kk}&{}=1\qquad {\text{for}}\ k\neq i,\,j\\g_{kk}&{}=c\qquad {\text{for}}\ k=i,\,j\\g_{ji}&{}=-g_{ij}=-s\\\end{aligned}}}$

The product G(i, j, θ)x represents a counterclockwise rotation of the vector x in the (i, j) plane of θ radians, hence the name Givens rotation.

Stable calculation

When a Givens rotation matrix, G(i, j, θ), multiplies another matrix, A, from the left, G A, only rows i and j of A are affected. Thus we restrict attention to the following counterclockwise problem. Given a and b, find c = cos θ and s = -sin θ such that

${\displaystyle {\begin{bmatrix}c&-s\\s&c\end{bmatrix}}{\begin{bmatrix}a\\b\end{bmatrix}}={\begin{bmatrix}r\\0\end{bmatrix}},}$

where ${\displaystyle r={\sqrt {a^{2}+b^{2}}}}$ is the length of the vector ${\displaystyle (a,b)}$. Explicit calculation of θ is rarely necessary or desirable. Instead we directly seek c and s. An obvious solution would be

${\displaystyle {\begin{aligned}c&{}\leftarrow a/r\\s&{}\leftarrow -b/r.\end{aligned}}}$^[1]

However, the computation for r may overflow or underflow. An alternative formulation avoiding this problem (Golub & Van Loan 1996, §5.1.8) is implemented as the hypot function in many programming languages.

The following Fortran code is a minimalistic implementation of Givens rotation for real numbers. If the input values 'a' or 'b' are frequently zero, the code may be optimized to handle these cases as presented here.

subroutine givens_rotation(a, b, c, s, r)

real a, b, c, s, r
real h, d

if (b .ne. 0.0) then
    h = hypot(a, b)
    d = 1.0 / h
    c = abs(a) * d
    s = sign(d, a) * b
    r = sign(1.0, a) * h
else
    c = 1.0
    s = 0.0
    r = a
end if

return
end

Furthermore, as Edward Anderson discovered in improving LAPACK, a previously overlooked numerical consideration is continuity. To achieve this, we require r to be positive.^[2] The following MATLAB/GNU Octave code illustrates the algorithm.

function [c, s, r] = givens_rotation(a, b)
    if b == 0;
        c = sign(a);
        if (c == 0);
            c = 1.0; % Unlike other languages, MatLab's sign function returns 0 on input 0.
        end;
        s = 0;
        r = abs(a);
    elseif a == 0;
        c = 0;
        s = -sign(b);
        r = abs(b);
    elseif abs(a) > abs(b);
        t = b / a;
        u = sign(a) * sqrt(1 + t * t);
        c = 1 / u;
        s = -c * t;
        r = a * u;
    else
        t = a / b;
        u = sign(b) * sqrt(1 + t * t);
        s = -1 / u;
        c = t / u;
        r = b * u;
    end
end

The IEEE 754 copysign(x,y) function, provides a safe and cheap way to copy the sign of y to x. If that is not available, |x|⋅sgn(y), using the abs and sgn functions, is an alternative as done above.

Triangularization

Given the following 3×3 Matrix:

${\displaystyle A_{1}={\begin{bmatrix}6&5&0\\5&1&4\\0&4&3\\\end{bmatrix}},}$

two iterations of the Givens rotation (note that the Givens rotation algorithm used here differs slightly from above) yield an upper triangular matrix in order to compute the QR decomposition.

In order to form the desired matrix, zeroing elements (2,1) and (3,2) is required; element (2,1) is zeroed first, using a rotation matrix of:

${\displaystyle G_{1}={\begin{bmatrix}c&-s&0\\s&c&0\\0&0&1\\\end{bmatrix}}.}$

The following matrix multiplication results:

${\displaystyle {\begin{aligned}G_{1}A_{1}&{}=A_{2}\\&{}={\begin{bmatrix}c&-s&0\\s&c&0\\0&0&1\\\end{bmatrix}}{\begin{bmatrix}6&5&0\\5&1&4\\0&4&3\\\end{bmatrix}},\end{aligned}}}$

where

${\displaystyle {\begin{aligned}r&{}={\sqrt {6^{2}+5^{2}}}\approx 7.8102\\c&{}=6/r\approx 0.7682\\s&{}=-5/r\approx -0.6402.\end{aligned}}}$

Using these values for c and s and performing the matrix multiplication above yields A₂:

${\displaystyle A_{2}\approx {\begin{bmatrix}7.8102&4.4813&2.5607\\0&-2.4327&3.0729\\0&4&3\\\end{bmatrix}}}$

Zeroing element (3,2) finishes off the process. Using the same idea as before, the rotation matrix is:

${\displaystyle G_{2}={\begin{bmatrix}1&0&0\\0&c&-s\\0&s&c\\\end{bmatrix}}}$

Afterwards, the following matrix multiplication is:

${\displaystyle {\begin{aligned}G_{2}A_{2}&{}=A_{3}\\&{}\approx {\begin{bmatrix}1&0&0\\0&c&-s\\0&s&c\\\end{bmatrix}}{\begin{bmatrix}7.8102&4.4813&2.5607\\0&-2.4327&3.0729\\0&4&3\\\end{bmatrix}},\end{aligned}}}$

where

${\displaystyle {\begin{aligned}r&{}\approx {\sqrt {(-2.4327)^{2}+4^{2}}}\approx 4.6817\\c&{}\approx -2.4327/r\approx -0.5196\\s&{}\approx -4/r\approx -0.8544.\end{aligned}}}$

Using these values for c and s and performing the multiplications results in A₃:

${\displaystyle A_{3}\approx {\begin{bmatrix}7.8102&4.4813&2.5607\\0&4.6817&0.9665\\0&0&-4.1843\\\end{bmatrix}}.}$

This new matrix A₃ is the upper triangular matrix needed to perform an iteration of the QR decomposition. Q is now formed using the transpose of the rotation matrices in the following manner:

${\displaystyle Q=G_{1}^{T}\,G_{2}^{T}.}$

Performing this matrix multiplication yields:

${\displaystyle Q\approx {\begin{bmatrix}0.7682&0.3327&0.5470\\0.6402&-0.3992&-0.6564\\0&0.8544&-0.5196\\\end{bmatrix}}.}$

This completes two iterations of the Givens Rotation and calculating the QR decomposition can now be done.

QR iteration variant

If performing the above calculations as a step in the QR algorithm for finding the eigenvalues of a matrix, then one next wants to compute the matrix ${\displaystyle RQ}$, but one should not do so by first multiplying ${\displaystyle G_{1}^{T}}$ and ${\displaystyle G_{2}^{T}}$ to form ${\displaystyle Q}$, instead rather by multiplying each ${\displaystyle G_{k}^{T}}$ by ${\displaystyle RG_{1}^{T}\dots G_{k-1}^{T}}$ (on the right). The reason for this is that each multiplication by a Givens matrix on the right changes only two columns of ${\displaystyle R}$, thus requiring a mere ${\displaystyle O(n)}$ arithmetic operations, which for ${\displaystyle n-1}$ Givens rotations sums up to ${\displaystyle O(n^{2})}$ arithmetic operations; multiplying by the general ${\displaystyle n\times n}$ matrix ${\displaystyle Q}$ would require ${\displaystyle O(n^{3})}$ arithmetic operations. Likewise, storing the full ${\displaystyle Q}$ matrix amounts to ${\displaystyle n^{2}}$ elements, but each Givens matrix is fully specified by its pair ${\displaystyle (c,s)}$ and ${\displaystyle n-1}$ of them can thus be stored in ${\displaystyle 2n-2}$ elements.

In the example, $${\displaystyle {\begin{aligned}RQ=A_{3}(G_{1}^{T}G_{2}^{T})={}&(A_{3}G_{1}^{T})G_{2}^{T}\\\approx {}&{\begin{bmatrix}8.8687&-1.5575&2.5607\\2.9972&3.5965&0.9665\\0&0&-4.1843\end{bmatrix}}G_{2}^{T}\approx {\begin{bmatrix}8.8687&2.9972&0.0\\2.9972&-1.0430&-3.5750\\0&-3.5750&2.1742\end{bmatrix}}\end{aligned}}}$$

Complex matrices

Another method can extend Givens rotations to complex matrices. A diagonal matrix whose diagonal elements have unit magnitudes but arbitrary phases is unitary. Let A be a matrix for which it is desired to make the ji element be zero using the rows and columns i and j>i. Let D be a diagonal matrix whose diagonal elements are one except the ii and jj diagonal elements which also have unit magnitude but have phases which are to be determined. The phases of the ii and jj elements of D can be chosen so as to make the ii and ji elements of the product matrix D A be real. Then a Givens rotation G can be chosen using the i and j>i rows and columns so as to make the ji element of the product matrix G D A be zero. Since a product of unitary matrices is unitary, the product matrix G D is unitary and so is any product of such matrix pair products.

In Clifford algebra

In Clifford algebra and its child structures such as geometric algebra, rotations are represented by bivectors. Givens rotations are represented by the exterior product of the basis vectors. Given any pair of basis vectors ${\displaystyle \mathbf {e} _{i},\mathbf {e} _{j}}$ Givens rotations bivectors are:

${\displaystyle B_{ij}=\mathbf {e} _{i}\wedge \mathbf {e} _{j}.}$

Their action on any vector is written:

${\displaystyle v=e^{-(\theta /2)(\mathbf {e} _{i}\wedge \mathbf {e} _{j})}ue^{(\theta /2)(\mathbf {e} _{i}\wedge \mathbf {e} _{j})},}$

where

${\displaystyle e^{(\theta /2)(\mathbf {e} _{i}\wedge \mathbf {e} _{j})}=\cos(\theta /2)+\sin(\theta /2)\mathbf {e} _{i}\wedge \mathbf {e} _{j}.}$

Dimension 3

See also: Euler angles and Davenport rotations

There are three Givens rotations in dimension 3:

${\displaystyle R_{X}(\theta )={\begin{bmatrix}1&0&0\\0&\cos \theta &-\sin \theta \\0&\sin \theta &\cos \theta \end{bmatrix}}.}$

${\displaystyle {\begin{aligned}\\R_{Y}(\theta )={\begin{bmatrix}\cos \theta &0&-\sin \theta \\0&1&0\\\sin \theta &0&\cos \theta \end{bmatrix}}\end{aligned}}}$^{[note 1]}

${\displaystyle {\begin{aligned}\\R_{Z}(\theta )={\begin{bmatrix}\cos \theta &-\sin \theta &0\\\sin \theta &\cos \theta &0\\0&0&1\end{bmatrix}}\end{aligned}}}$

Given that they are endomorphisms they can be composed with each other as many times as desired, keeping in mind that g ∘ f ≠ f ∘ g.

These three Givens rotations composed can generate any rotation matrix according to Davenport's chained rotation theorem. This means that they can transform the standard basis of the space to any other frame in the space.^{[clarification needed]}

When rotations are performed in the right order, the values of the rotation angles of the final frame will be equal to the Euler angles of the final frame in the corresponding convention. For example, an operator ${\displaystyle R=R_{Y}(\theta _{3})\cdot R_{X}(\theta _{2})\cdot R_{Z}(\theta _{1})}$ transforms the basis of the space into a frame with angles roll, pitch and yaw ${\displaystyle YPR=(\theta _{3},\theta _{2},\theta _{1})}$ in the Tait–Bryan convention z-x-y (convention in which the line of nodes is perpendicular to z and Y axes, also named Y-X′-Z″).

For the same reason, any rotation matrix in 3D can be decomposed in a product of three of these rotation operators.

The meaning of the composition of two Givens rotations g ∘ f is an operator that transforms vectors first by f and then by g, being f and g rotations about one axis of basis of the space. This is similar to the extrinsic rotation equivalence for Euler angles.

Table of composed rotations

The following table shows the three Givens rotations equivalent to the different Euler angles conventions using extrinsic composition (composition of rotations about the basis axes) of active rotations and the right-handed rule for the positive sign of the angles.

The notation has been simplified in such a way that c₁ means cos θ₁ and s₂ means sin θ₂). The subindexes of the angles are the order in which they are applied using extrinsic composition (1 for intrinsic rotation, 2 for nutation, 3 for precession)

As rotations are applied just in the opposite order of the Euler angles table of rotations, this table is the same but swapping indexes 1 and 3 in the angles associated with the corresponding entry. An entry like zxy means to apply first the y rotation, then x, and finally z, in the basis axes.

All the compositions assume the right hand convention for the matrices that are multiplied, yielding the following results.

xzx	${\displaystyle {\begin{bmatrix}c_{2}&-c_{1}s_{2}&s_{1}s_{2}\\c_{3}s_{2}&c_{3}c_{2}c_{1}-s_{3}s_{1}&-c_{2}c_{3}s_{1}-c_{1}s_{3}\\s_{2}s_{3}&c_{3}s_{1}+c_{1}c_{2}s_{3}&c_{3}c_{1}-c_{2}s_{3}s_{1}\end{bmatrix}}}$	xzy	${\displaystyle {\begin{bmatrix}c_{2}c_{3}&-c_{3}s_{2}c_{1}+s_{3}s_{1}&c_{3}s_{2}s_{1}+s_{3}c_{1}\\s_{2}&c_{1}c_{2}&-c_{2}s_{1}\\-s_{3}c_{2}&s_{3}s_{2}c_{1}+c_{3}s_{1}&-s_{3}s_{2}s_{1}+c_{3}c_{1}\end{bmatrix}}}$
xyx	${\displaystyle {\begin{bmatrix}c_{2}&s_{1}s_{2}&c_{1}s_{2}\\s_{2}s_{3}&c_{3}c_{1}-c_{2}s_{3}s_{1}&-c_{3}s_{1}-c_{1}c_{2}s_{3}\\-c_{3}s_{2}&c_{3}c_{2}s_{1}+c_{1}s_{3}&c_{3}c_{2}c_{1}-s_{3}s_{1}\end{bmatrix}}}$	xyz	${\displaystyle {\begin{bmatrix}c_{3}c_{2}&-s_{3}c_{1}+c_{3}s_{2}s_{1}&s_{3}s_{1}+c_{3}s_{2}c_{1}\\s_{3}c_{2}&c_{3}c_{1}+s_{3}s_{2}s_{1}&-c_{3}s_{1}+s_{3}s_{2}c_{1}\\-s_{2}&c_{2}s_{1}&c_{2}c_{1}\end{bmatrix}}}$
yxy	${\displaystyle {\begin{bmatrix}c_{3}c_{1}-c_{2}s_{3}s_{1}&s_{2}s_{3}&c_{3}s_{1}+s_{3}c_{2}c_{1}\\s_{1}s_{2}&c_{2}&-c_{1}s_{2}\\-c_{2}c_{3}s_{1}-c_{1}s_{3}&c_{3}s_{2}&c_{3}c_{2}c_{1}-s_{3}s_{1}\end{bmatrix}}}$	yxz	${\displaystyle {\begin{bmatrix}c_{3}c_{1}-s_{3}s_{2}s_{1}&-s_{3}c_{2}&c_{3}s_{1}+s_{3}s_{2}c_{1}\\s_{3}c_{1}+c_{3}s_{2}s_{1}&c_{3}c_{2}&s_{3}s_{1}-c_{3}s_{2}c_{1}\\-c_{2}s_{1}&s_{2}&c_{2}c_{1}\end{bmatrix}}}$
yzy	${\displaystyle {\begin{bmatrix}c_{3}c_{2}c_{1}-s_{3}s_{1}&-c_{3}s_{2}&c_{2}c_{3}s_{1}+c_{1}s_{3}\\c_{1}s_{2}&c_{2}&s_{1}s_{2}\\-c_{3}s_{1}-c_{1}c_{2}s_{3}&s_{2}s_{3}&c_{3}c_{1}-c_{2}s_{3}s_{1}\end{bmatrix}}}$	yzx	${\displaystyle {\begin{bmatrix}c_{2}c_{1}&-s_{2}&c_{2}s_{1}\\c_{3}s_{2}c_{1}+s_{3}s_{1}&c_{3}c_{2}&c_{3}s_{2}s_{1}-s_{3}c_{1}\\s_{3}s_{2}c_{1}-c_{3}s_{1}&s_{3}c_{2}&s_{3}s_{2}s_{1}+c_{3}c_{1}\end{bmatrix}}}$
zyz	${\displaystyle {\begin{bmatrix}c_{3}c_{2}c_{1}-s_{3}s_{1}&-c_{2}s_{1}c_{3}-c_{1}s_{3}&c_{3}s_{2}\\c_{3}s_{1}+c_{1}c_{2}s_{3}&c_{3}c_{1}-c_{2}s_{3}s_{1}&s_{2}s_{3}\\-c_{1}s_{2}&s_{1}s_{2}&c_{2}\end{bmatrix}}}$	zyx	${\displaystyle {\begin{bmatrix}c_{2}c_{1}&-c_{2}s_{1}&s_{2}\\s_{3}s_{2}c_{1}+c_{3}s_{1}&-s_{3}s_{2}s_{1}+c_{3}c_{1}&-s_{3}c_{2}\\-c_{3}s_{2}c_{1}+s_{3}s_{1}&c_{3}s_{2}s_{1}+s_{3}c_{1}&c_{3}c_{2}\end{bmatrix}}}$
zxz	${\displaystyle {\begin{bmatrix}c_{3}c_{1}-c_{2}s_{1}s_{3}&-c_{3}s_{1}-c_{1}c_{2}s_{3}&s_{2}s_{3}\\c_{2}c_{3}s_{1}+c_{1}s_{3}&c_{3}c_{2}c_{1}-s_{3}s_{1}&-c_{3}s_{2}\\s_{1}s_{2}&c_{1}s_{2}&c_{2}\end{bmatrix}}}$	zxy	${\displaystyle {\begin{bmatrix}c_{3}c_{1}+s_{3}s_{2}s_{1}&-c_{3}s_{1}+s_{3}s_{2}c_{1}&s_{3}c_{2}\\c_{2}s_{1}&c_{2}c_{1}&-s_{2}\\-s_{3}c_{1}+c_{3}s_{2}s_{1}&s_{3}s_{1}+c_{3}s_{2}c_{1}&c_{3}c_{2}\end{bmatrix}}}$

See also

• Jacobi rotation
• Plane of rotation
• Householder transformation
• Davenport rotations

Notes

1. The ${\displaystyle R_{Y}(\theta )}$ rotation matrix immediately below is not a Givens rotation. The ${\displaystyle R_{Y}(\theta )}$ matrix immediately below respects the right-hand rule and is this usual matrix one sees in Computer Graphics; however, a Givens rotation is simply a matrix as defined in the Matrix representation section above and does not necessarily respect the right-hand rule. The matrix below is actually the Givens rotation through an angle of -${\displaystyle \theta }$.

   ${\displaystyle R_{Y}(\theta )={\begin{bmatrix}\cos \theta &0&\sin \theta \\0&1&0\\-\sin \theta &0&\cos \theta \end{bmatrix}}}$

Citations

1. Björck, Ake (1996). Numerical Methods for Least Squares Problems. United States: SIAM. p. 54. ISBN 9780898713602. Retrieved 16 August 2016.
2. Anderson, Edward (4 December 2000). "Discontinuous Plane Rotations and the Symmetric Eigenvalue Problem" (PDF). LAPACK Working Note. University of Tennessee at Knoxville and Oak Ridge National Laboratory. Retrieved 16 August 2016.

References

• Bindel, D.; Demmel, J.; Kahan, W.; Marques, O. (2000), On Computing Givens rotations reliably and efficiently. LAPACK Working Note 148, University of Tennessee, UT-CS-00-449, January 31, 2001.
• Cybenko, George (March–April 2001), "Reducing Quantum Computations to Elementary Unitary Operations" (PDF), Computing in Science and Engineering, 3 (2): 27–32, doi:10.1109/5992.908999, archived from the original (PDF) on 2016-03-03, retrieved 2009-02-26
• Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Johns Hopkins, ISBN 978-0-8018-5414-9.
• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 11.3.1. Givens Method", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8, archived from the original on 2011-08-11, retrieved 2011-08-13