Editing Givens rotation

{{Short description|Concept in numerical linear algebra}}
In [[numerical linear algebra]], a '''Givens rotation''' is a [[Rotation (mathematics)|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 matrix#Row-addition_transformations|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 <math>x_1</math> through <math>x_n</math> and the elements of the other target row by <math>y_1</math> through <math>y_n</math>:
<math display="block">
  \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}
</math>
Then the effect of a Givens rotation is to rotate each subvector <math>(x_k,y_k)</math> 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 <math>(x_k,y_k)</math> by the same angle, but here these named elements occur in the matrix as
<math display="block">
  \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}
</math>
Some algorithms, especially those concerned with preserving [[matrix similarity]], apply Givens rotations as a [[Conjugacy_class#Conjugacy_as_group_action|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 transformation]]s 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 (mathematics)|matrix]] of the form

:<math>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},</math>

where {{math|1=''c'' = cos ''θ''}} and {{math|1=''s'' = sin ''θ''}} appear at the intersections {{mvar|i}}th and {{mvar|j}}th rows and columns. That is, for fixed {{mvar|i}} {{mvar|>}} {{mvar|j}}, the non-zero elements of Givens matrix are given by:
:<math>\begin{align}
 g_{kk} &{}= 1 \qquad \text{for} \ k \ne i,\,j\\
 g_{kk} &{}= c \qquad \text{for} \ k = i,\,j\\
 g_{ji} &{} = -g_{ij}= -s\\
\end{align}</math>

The product {{math|''G''(''i'', ''j'', ''θ'')'''x'''}} represents a counterclockwise rotation of the [[Euclidean vector|vector]] {{math|'''x'''}} in the {{math|(''i'', ''j'')}} plane of {{mvar|θ}} radians, hence the name Givens rotation.

== Stable calculation ==
When a Givens rotation matrix, {{math|''G''(''i'', ''j'', ''θ'')}}, multiplies another matrix, {{mvar|A}}, from the left, {{math|''G A''}}, only rows {{mvar|i}} and {{mvar|j}} of {{mvar|A}} are affected. Thus we restrict attention to the following counterclockwise problem. Given {{mvar|a}} and {{mvar|b}}, find {{math|1=''c'' = cos ''θ''}} and {{math|1=''s'' = sin ''θ''}} such that
:<math> \begin{bmatrix} c & -s \\ s & c \end{bmatrix} \begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} r \\ 0 \end{bmatrix} , </math>
where <math> r = \sqrt{a^2 + b^2} </math> is the length of the vector <math>(a,b)</math>.
Explicit calculation of {{mvar|θ}} is rarely necessary or desirable. Instead we directly seek {{mvar|c}} and {{mvar|s}}. An obvious solution would be
:<math>\begin{align}
 c &{}\larr a / r \\
 s &{}\larr -b / r.
\end{align}</math><ref>{{cite book|last1=Björck|first1=Ake|title=Numerical Methods for Least Squares Problems|date=1996|publisher=SIAM|location=United States|isbn=9780898713602|page=54|url=https://books.google.com/books?id=aQD1LLYz6tkC|accessdate=16 August 2016|language=en}}</ref>

However, the computation for {{mvar|r}} may [[arithmetic overflow|overflow]] or underflow. An alternative formulation avoiding this problem {{harv|Golub|Van Loan|1996|loc=§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 [https://dl.acm.org/citation.cfm?doid=567806.567809 here].

<syntaxhighlight lang=fortran>
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
</syntaxhighlight>


Furthermore, as Edward Anderson discovered in improving [[LAPACK]], a previously overlooked numerical consideration is continuity. To achieve this, we require {{mvar|r}} to be positive.<ref>{{cite web|publisher=University of Tennessee at Knoxville and Oak Ridge National Laboratory|last1=Anderson|first1=Edward|title=Discontinuous Plane Rotations and the Symmetric Eigenvalue Problem|url=http://www.netlib.org/lapack/lawnspdf/lawn150.pdf|accessdate=16 August 2016|date=4 December 2000|series=LAPACK Working Note}}</ref> The following [[MATLAB]]/[[GNU Octave]] code illustrates the algorithm. 

<syntaxhighlight lang="matlab">
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
</syntaxhighlight>

The [[IEEE 754]] <code>copysign(x,y)</code> function, provides a safe and cheap way to copy the sign of <code>y</code> to <code>x</code>. If that is not available, {{math|{{abs|''x''}}⋅sgn(''y'')}}, using the [[absolute value|abs]] and [[sign function|sgn]] functions, is an alternative as done above.

== Triangularization ==
Given the following {{gaps|3|×|3}} Matrix:

:<math> A_1 =
       \begin{bmatrix}   6    &    5    &    0   \\
                         5    &    1    &    4     \\
                         0    &    4    &    3     \\
       \end{bmatrix},</math>

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 {{gaps|(2,|1)}} and {{gaps|(3,|2)}} is required; element {{gaps|(2,|1)}} is zeroed first, using a rotation matrix of:
:<math>G_{1} =
       \begin{bmatrix}   c    &    -s    &    0   \\
                         s   &    c    &    0     \\
                         0    &    0    &    1     \\
       \end{bmatrix}.</math>

The following matrix multiplication results:
:<math>
\begin{align}
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{align}
</math>

where
:<math>\begin{align}
 r &{}= \sqrt{6^2 + 5^2} \approx 7.8102 \\
 c &{}= 6 / r \approx 0.7682 \\
 s &{}= -5 / r \approx -0.6402.
\end{align}
</math>

Using these values for {{mvar|c}} and {{mvar|s}} and performing the matrix multiplication above yields {{mvar|A<sub>2</sub>}}:
:<math>A_2 \approx \begin{bmatrix}   7.8102    &     4.4813    &    2.5607   \\
                                 0    &    -2.4327    &    3.0729   \\
                                 0    &          4    &         3   \\
       \end{bmatrix}</math>

Zeroing element {{gaps|(3,|2)}} finishes off the process. Using the same idea as before, the rotation matrix is:
:<math>G_{2} =
       \begin{bmatrix}   1    &    0    &    0   \\
                         0   &    c    &    -s     \\
                         0    &    s   &    c     \\
       \end{bmatrix}</math>

Afterwards, the following matrix multiplication is:
:<math>
\begin{align}
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{align}
</math>

where
:<math>\begin{align}
 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{align}
</math>
Using these values for {{mvar|c}} and {{mvar|s}} and performing the multiplications results in {{mvar|A<sub>3</sub>}}:
:<math>A_3 \approx
       \begin{bmatrix}   7.8102    &    4.4813    &    2.5607   \\
                         0   &    4.6817    &    0.9665     \\
                         0    &    0    &    -4.1843     \\
       \end{bmatrix}.
</math>

This new matrix {{mvar|A<sub>3</sub>}} is the upper triangular matrix needed to perform an iteration of the [[QR decomposition]].  {{mvar|Q}} is now formed using the transpose of the rotation matrices in the following manner:

:<math>Q = G_{1}^T\, G_{2}^T.
</math>

Performing this matrix multiplication yields:
:<math>Q \approx
       \begin{bmatrix}   0.7682    &   0.3327    &    0.5470   \\
                         0.6402   &     -0.3992    &    -0.6564     \\
                         0    &    0.8544    &     -0.5196     \\
       \end{bmatrix}.</math>

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 <math> RQ </math>, but one should not do so by first multiplying <math> G_{1}^T </math> and <math> G_{2}^T </math> to form <math> Q </math>, instead rather by multiplying each <math> G_k^T </math> by <math> R G_1^T \dots G_{k-1}^T </math> (on the right). The reason for this is that each multiplication by a Givens matrix on the right changes only two columns of <math> R </math>, thus requiring a mere <math> O(n) </math> arithmetic operations, which for <math> n-1 </math> Givens rotations sums up to <math> O(n^2) </math> arithmetic operations; multiplying by the general <math> n \times n </math> matrix <math>Q</math> would require <math> O(n^3) </math> arithmetic operations. Likewise, storing the full <math>Q</math> matrix amounts to <math>n^2</math> elements, but each Givens matrix is fully specified by its pair <math>(c,s)</math> and <math> n-1 </math> of them can thus be stored in <math> 2n-2 </math> elements.

In the example,
<math display="block">
  \begin{aligned}
    R Q = 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}
</math>

==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 algebras==

In [[Clifford algebra]]s and its child structures such as [[geometric algebra]]s, rotations are represented by [[bivector]]s. Givens rotations are represented by the exterior product of the basis vectors. Given any pair of basis vectors <math>\mathbf e_i, \mathbf e_j</math> Givens rotations bivectors are:

: <math>B_{ij} = \mathbf e_i \wedge \mathbf e_j.</math>

Their action on any vector is written:

: <math>v=e^{-(\theta/2)(\mathbf e_i \wedge \mathbf e_j)}u e^{(\theta/2)(\mathbf e_i \wedge \mathbf e_j)},</math>

where

: <math>e^{(\theta/2)(\mathbf e_i \wedge \mathbf e_j)}= \cos(\theta/2)+  \sin(\theta/2) \mathbf e_i \wedge \mathbf e_j.</math>

==Dimension 3==
{{see also|Euler angles|Davenport rotations}}

There are three Givens rotations in dimension 3:

:<math>
R_X(\theta) =
\begin{bmatrix}
1 & 0 & 0 \\
0 & \cos \theta & -\sin \theta \\
0 & \sin \theta & \cos \theta
\end{bmatrix}. 
</math>

:<math>\begin{align} \\
R_Y(\theta) =
\begin{bmatrix}
\cos \theta & 0 & -\sin \theta \\
0 & 1 & 0  \\
\sin \theta & 0 & \cos \theta
\end{bmatrix}
\end{align}
</math>{{#tag:ref|The <math>R_Y(\theta)</math> rotation matrix immediately below is ''not'' a Givens rotation.  The <math>R_Y(\theta)</math> 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|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 -<math>\theta</math>.

:<math>
R_Y(\theta) =
\begin{bmatrix}
\cos \theta & 0 & \sin \theta \\
0 & 1 & 0  \\
-\sin \theta & 0 & \cos \theta
\end{bmatrix}
</math>
|group=note}}

:<math>\begin{align} \\
R_Z(\theta) =
\begin{bmatrix}
\cos \theta & -\sin \theta & 0 \\
\sin \theta & \cos \theta & 0 \\
0 & 0 & 1 
\end{bmatrix}
\end{align}
</math>

Given that they are [[endomorphism]]s they can be composed with each other as many times as desired, keeping in mind that {{math|''g'' ∘ ''f'' ≠ ''f'' ∘ ''g''}}.

These three Givens rotations [[Function composition#Rotation composition|composed]] can generate any rotation matrix according to [[Davenport chained rotations|Davenport's chained rotation theorem]]. This means that they can [[transformation (geometry)|transform]] the [[standard basis]] of the space to any other frame in the space.{{clarify|date=March 2014}}

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 <math>R = R_Y(\theta_3)\cdot R_X(\theta_2)\cdot R_Z(\theta_1)</math> transforms the basis of the space into a frame with angles roll, pitch and yaw <math>YPR = (\theta_3,\theta_2,\theta_1)</math> in the [[Tait–Bryan angles|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 [[three-dimensional rotation operator|rotation operators]].

The meaning of the composition of two Givens rotations {{math|''g'' ∘ ''f''}} is an operator that transforms vectors first by {{mvar|f}} and then by {{mvar|g}}, being {{mvar|f}} and {{mvar|g}} rotations about one axis of basis of the space. This is similar to the [[Euler angles#Euler angles as composition of extrinsic rotations|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 and passive transformation|active rotations]] and the right-handed rule for the positive sign of the angles.

The notation has been simplified in such a way that {{math|''c''<sub>1</sub>}} means {{math|cos ''θ''<sub>1</sub>}} and {{math|''s''<sub>2</sub>}} means {{math|sin ''θ''<sub>2</sub>)}}. 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 composed rotations|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.

:{| class="wikitable" style="background-color:white; font-weight:500"
|-
!''xzx''
|<math>\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}</math>
!''xzy''
|<math>\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}</math>
|-
!''xyx''
|<math>\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}</math>
!''xyz''
|<math>\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}</math>
|-
!''yxy''
|<math>\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}</math>
!''yxz''
|<math>\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}</math>
|-
!''yzy''
|<math>\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}</math>
!''yzx''
|<math>\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}</math>
|-
!''zyz''
|<math>\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}</math>
!''zyx''
|<math>\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}</math>
|-
!''zxz''
|<math>\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}</math>
!''zxy''
|<math>\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}</math>
|}

== See also ==
* [[Jacobi rotation]]
* [[Plane of rotation]]
* [[Householder transformation]]
* [[Davenport rotations]]

==Notes==
{{reflist|group=note}}

==Citations==
{{reflist}}

==References==
* {{Citation
  | last1=Bindel | first1=D.
  | last2=Demmel | first2=J. | author2-link=James Demmel
  | last3=Kahan | first3=W. | author3-link=William Kahan
  | last4=Marques | first4=O.
  | url=http://www.netlib.org/lapack/lawns/downloads/
  | title=On Computing Givens rotations reliably and efficiently
  | year=2000 }}. LAPACK Working Note 148, University of Tennessee, UT-CS-00-449, January 31, 2001.<!-- The paper itself says January 2001, but the LAWN site says October 2000. -->
* {{Citation
  | author-link1 = George Cybenko
  | last = Cybenko
  | first = George
  | title = Reducing Quantum Computations to Elementary Unitary Operations
  | journal = Computing in Science and Engineering
  | volume = 3
  | issue = 2
  | pages = 27–32
  | date = March–April 2001
  | doi = 10.1109/5992.908999
  | url = http://vlsicad.eecs.umich.edu/Quantum/papers/QC-Unitary.pdf
  | access-date = 2009-02-26
  | archive-date = 2016-03-03
  | archive-url = https://web.archive.org/web/20160303213109/http://vlsicad.eecs.umich.edu/Quantum/papers/QC-Unitary.pdf
  | url-status = dead
  }}
* {{Citation | last1=Golub | first1=Gene H. | author1-link=Gene H. Golub | last2=Van Loan | first2=Charles F. | author2-link=Charles F. Van Loan | title=Matrix Computations | publisher=Johns Hopkins | edition=3rd | isbn=978-0-8018-5414-9 | year=1996}}.
*{{Citation | last1=Press | first1=WH | author1-link=William H. Press | last2=Teukolsky | first2=SA | author2-link=Saul Teukolsky | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | author4-link=Brian P. Flannery | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | location=New York | isbn=978-0-521-88068-8 | chapter=Section 11.3.1. Givens Method | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=578 | access-date=2011-08-13 | archive-date=2011-08-11 | archive-url=https://web.archive.org/web/20110811154417/http://apps.nrbook.com/empanel/index.html#pg=578 | url-status=dead }}

{{Numerical linear algebra}}

[[Category:Articles with example MATLAB/Octave code]]
[[Category:Matrices (mathematics)]]
[[Category:Numerical linear algebra]]
[[Category:Rotation in three dimensions]]