Right Qr 1 2

In numerical linear algebra, the QR algorithm or QR iteration is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix. The QR algorithm was developed in the late 1950s by John G. F. Francis and by Vera N. Kublanovskaya, working independently.^[1][2][3] 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.

The practical QR algorithm[edit]

Formally, let A be a real matrix of which we want to compute the eigenvalues, and let A₀:=A. At the k-th step (starting with k = 0), we compute the QR decompositionA_k=Q_kR_k where Q_k is an orthogonal matrix (i.e., Q^T = Q⁻¹) and R_k is an upper triangular matrix. We then form A_k+1 = R_kQ_k. Note that

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${\displaystyle A_{k+1}=R_k{Q}_k=Q_k^{-1}{Q}_k{R}_k{Q}_k=Q_k^{-1}{A}_k{Q}_k=Q_k^{\mathsf{T}}{A}_k{Q}_k,}$

so all the A_k are similar and hence they have the same eigenvalues. The algorithm is numerically stable because it proceeds by orthogonal similarity transforms.

Under certain conditions,^[4] the matrices A_k converge to a triangular matrix, the Schur form of A. The eigenvalues of a triangular matrix are listed on the diagonal, and the eigenvalue problem is solved. In testing for convergence it is impractical to require exact zeros,^{[citation needed]} but the Gershgorin circle theorem provides a bound on the error.

In this crude form the iterations are relatively expensive. This can be mitigated by first bringing the matrix A to upper Hessenberg form (which costs ${\displaystyle \begin{array}{c}\frac{10}{3}\end{array}{n}^3+\mathcal{O}(n^2)}$ arithmetic operations using a technique based on Householder reduction), with a finite sequence of orthogonal similarity transforms, somewhat like a two-sided QR decomposition.^[5][6] (For QR decomposition, the Householder reflectors are multiplied only on the left, but for the Hessenberg case they are multiplied on both left and right.) Determining the QR decomposition of an upper Hessenberg matrix costs ${\displaystyle 6n^2+\mathcal{O}(n)}$ arithmetic operations. Moreover, because the Hessenberg form is already nearly upper-triangular (it has just one nonzero entry below each diagonal), using it as a starting point reduces the number of steps required for convergence of the QR algorithm.

If the original matrix is symmetric, then the upper Hessenberg matrix is also symmetric and thus tridiagonal, and so are all the A_k. This procedure costs ${\displaystyle \begin{array}{c}\frac{4}{3}\end{array}{n}^3+\mathcal{O}(n^2)}$ arithmetic operations using a technique based on Householder reduction.^[5][6] Determining the QR decomposition of a symmetric tridiagonal matrix costs ${\displaystyle \mathcal{O}(n)}$ operations.^[7]

The rate of convergence depends on the separation between eigenvalues, so a practical algorithm will use shifts, either explicit or implicit, to increase separation and accelerate convergence. A typical symmetric QR algorithm isolates each eigenvalue (then reduces the size of the matrix) with only one or two iterations, making it efficient as well as robust.^{[clarification needed]}

Ex 6.3, 4 In figure, 𝑄𝑅/𝑄𝑆 = 𝑄𝑇/𝑃𝑅 and ∠1 = ∠2. Show that ΔPQS ∼ ΔTQR. Given: 𝑄𝑅/𝑄𝑆=𝑄𝑇/𝑃𝑅 ∠ 1= ∠ 2 To prove: ΔPQS ∼ ΔTQR Proof: Given ∠ 1 = ∠ 2 PR = QP Given 𝑄𝑅/𝑄𝑆=𝑄𝑇/𝑃𝑅 Putting (1) 𝑄𝑅/𝑄𝑆=𝑄𝑇/𝑄𝑃 In ΔPQS and ΔTQR ∠ 𝑃𝑄𝑆= ∠ 𝑇𝑄𝑅 & 𝑄𝑅/𝑄𝑆=𝑄𝑇. To create a PDF QR Code, select Create QR Code on the Dashboard and choose PDF. Select a PDF up to 20 MB from your computer and upload it in the document field. If you want to simply link right to the PDF, tick the 'Link directly' box. NCERT Solutions Class 10 Maths Chapter 8 Introduction to Trigonometry have been provided here to help students prepare for the 2020-2021 CBSE exams. These solutions are prepared by the subject experts at BYJU'S as per the updated CBSE Syllabus (2020-2021).

The implicit QR algorithm[edit]

In modern computational practice, the QR algorithm is performed in an implicit version which makes the use of multiple shifts easier to introduce.^[4] The matrix is first brought to upper Hessenberg form ${\displaystyle A_0=QA{Q}^{\mathsf{T}}}$ as in the explicit version; then, at each step, the first column of ${\displaystyle A_k}$ is transformed via a small-size Householder similarity transformation to the first column of ${\displaystyle p(A_k)}$ (or ${\displaystyle p(A_k)e_1}$), where ${\displaystyle p(A_k)}$, of degree ${\displaystyle r}$, is the polynomial that defines the shifting strategy (often ${\displaystyle p(x)=(x-\lambda )(x-\overline{\lambda })}$, where ${\displaystyle \lambda }$ and ${\displaystyle \overline{\lambda }}$ are the two eigenvalues of the trailing ${\displaystyle 2\times 2}$ principal submatrix of ${\displaystyle A_k}$, the so-called implicit double-shift). Then successive Householder transformations of size ${\displaystyle r+1}$ are performed in order to return the working matrix ${\displaystyle A_k}$ to upper Hessenberg form. This operation is known as bulge chasing, due to the peculiar shape of the non-zero entries of the matrix along the steps of the algorithm. As in the first version, deflation is performed as soon as one of the sub-diagonal entries of ${\displaystyle A_k}$ is sufficiently small.

Renaming proposal[edit]

Since in the modern implicit version of the procedure no QR decompositions are explicitly performed, some authors, for instance Watkins,^[8] suggested changing its name to Francis algorithm. Golub and Van Loan use the term Francis QR step.

Interpretation and convergence[edit]

The QR algorithm can be seen as a more sophisticated variation of the basic 'power' eigenvalue algorithm. Recall that the power algorithm repeatedly multiplies A times a single vector, normalizing after each iteration. The vector converges to an eigenvector of the largest eigenvalue. Instead, the QR algorithm works with a complete basis of vectors, using QR decomposition to renormalize (and orthogonalize). For a symmetric matrix A, upon convergence, AQ = QΛ, where Λ is the diagonal matrix of eigenvalues to which A converged, and where Q is a composite of all the orthogonal similarity transforms required to get there. Thus the columns of Q are the eigenvectors.

History[edit]

The QR algorithm was preceded by the LR algorithm, which uses the LU decomposition instead of the QR decomposition. The QR algorithm is more stable, so the LR algorithm is rarely used nowadays. However, it represents an important step in the development of the QR algorithm.

The LR algorithm was developed in the early 1950s by Heinz Rutishauser, who worked at that time as a research assistant of Eduard Stiefel at ETH Zurich. Stiefel suggested that Rutishauser use the sequence of moments y₀^TA^kx₀, k = 0, 1, … (where x₀ and y₀ are arbitrary vectors) to find the eigenvalues of A. Rutishauser took an algorithm of Alexander Aitken for this task and developed it into the quotient–difference algorithm or qd algorithm. After arranging the computation in a suitable shape, he discovered that the qd algorithm is in fact the iteration A_k = L_kU_k (LU decomposition), A_k+1 = U_kL_k, applied on a tridiagonal matrix, from which the LR algorithm follows.^[9]

Other variants[edit]

One variant of the QR algorithm, the Golub-Kahan-Reinsch algorithm starts with reducing a general matrix into a bidiagonal one.^[10] This variant of the QR algorithm for the computation of singular values was first described by Golub & Kahan (1965) harvtxt error: no target: CITEREFGolubKahan1965 (help). The LAPACK subroutine DBDSQR implements this iterative method, with some modifications to cover the case where the singular values are very small (Demmel & Kahan 1990) harv error: no target: CITEREFDemmelKahan1990 (help). Together with a first step using Householder reflections and, if appropriate, QR decomposition, this forms the DGESVD routine for the computation of the singular value decomposition. The QR algorithm can also be implemented in infinite dimensions with corresponding convergence results.^[11][12]

References[edit]

1. ^J.G.F. Francis, 'The QR Transformation, I', The Computer Journal, 4(3), pages 265–271 (1961, received October 1959). doi:10.1093/comjnl/4.3.265
2. ^Francis, J. G. F. (1962). 'The QR Transformation, II'. The Computer Journal. 4 (4): 332–345. doi:10.1093/comjnl/4.4.332.
3. ^Vera N. Kublanovskaya, 'On some algorithms for the solution of the complete eigenvalue problem,' USSR Computational Mathematics and Mathematical Physics, vol. 1, no. 3, pages 637–657 (1963, received Feb 1961). Also published in: Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, vol.1, no. 4, pages 555–570 (1961). doi:10.1016/0041-5553(63)90168-X
4. ^ ^abGolub, G. H.; Van Loan, C. F. (1996). Matrix Computations (3rd ed.). Baltimore: Johns Hopkins University Press. ISBN0-8018-5414-8.
5. ^ ^abDemmel, James W. (1997). Applied Numerical Linear Algebra. SIAM.
6. ^ ^abTrefethen, Lloyd N.; Bau, David (1997). Numerical Linear Algebra. SIAM.
7. ^Ortega, James M.; Kaiser, Henry F. (1963). 'The LL^T and QR methods for symmetric tridiagonal matrices'. The Computer Journal. 6 (1): 99–101. doi:10.1093/comjnl/6.1.99.
8. ^Watkins, David S. (2007). The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods. Philadelphia, PA: SIAM. ISBN978-0-89871-641-2.
9. ^Parlett, Beresford N.; Gutknecht, Martin H. (2011), 'From qd to LR, or, how were the qd and LR algorithms discovered?'(PDF), IMA Journal of Numerical Analysis, 31 (3): 741–754, doi:10.1093/imanum/drq003, hdl:20.500.11850/159536, ISSN0272-4979
10. ^Bochkanov Sergey Anatolyevich. ALGLIB User Guide - General Matrix operations - Singular value decomposition . ALGLIB Project. 2010-12-11. URL:http://www.alglib.net/matrixops/general/svd.php.^{[permanent dead link]} Accessed: 2010-12-11. (Archived by WebCite at https://www.webcitation.org/5utO4iSnR?url=http://www.alglib.net/matrixops/general/svd.php
11. ^Deift, Percy; Li, Luenchau C.; Tomei, Carlos (1985). 'Toda flows with infinitely many variables'. Journal of Functional Analysis. 64 (3): 358–402. doi:10.1016/0022-1236(85)90065-5.
12. ^Colbrook, Matthew J.; Hansen, Anders C. (2019). 'On the infinite-dimensional QR algorithm'. Numerische Mathematik. 143 (1): 17–83. doi:10.1007/s00211-019-01047-5.

External links[edit]

• 'Eigenvalue problem'. PlanetMath.
• Notes on orthogonal bases and the workings of the QR algorithm by Peter J. Olver

Structure of Barcode

The following explains the format of a barcode.

Traxxas Qr-1

Quiet zone (margin)

The right and left ends of the barcode symbol.
If the margin is not wide enough, the barcode reader cannot scan the barcode data.
Both the right and left margins should be at least 10 times as wide as the narrow bar width (minimum element width). (See page here for narrow bar width.)

One Point

Unless the quiet zones are wide enough, it becomes unstable for the barcode reader to scan the barcode data.

Start/stop character

Character to indicate start or end of the data.
Start/stop character varies depending on the type of the barcode. CODE 39 uses “*” and CODABAR uses “a”,“b”,“c” and “d”. (EAN and ITF use, instead of a character, a bar pattern which indicates the start/end of the data.)

Data (message)

Bar patterns representing the data (numerical characters, alphabet, etc.) are arranged from left.
The drawing above shows that the data of “012” is demonstrated, with the bar patterns representing 0, 1, and 2 being arranged respectively from the left.

Check digit

A numeric value calculated to check for read error. It is appended directly after the barcode data.
(See page here for details)

Barcode length

Barcode length defines the total length including the right and left quiet zones.
If the barcode length, includinng quiet zones, does not fit in the scan width, the barcode reader cannot scan the data.

Barcode height

It is recommended to make the barcode as high as the printer permits.
If the barcode is not high enough, the laser may deviate from the barcode, causing unstable readings.
It is recommended to keep the height greater than 15 % of the barcode length.

Narrow Bar and Wide Bar

The following explains bars and spaces that are the minimum unit to form a barcode. A barcode is a combination of narrow and wide bars and spaces. Each bar and space are named as listed below:

Narrow and wide widths are determined at the following ratio:
NB:WB = NS:WS =1:2 to 1:3

One Point

If the ratio of narrow and wide is out of the range above, the barcode reader may perform unstable readings.
For preparing a barcode, careful attention shall be made to this ratio. The following ratio is normally recommended for preparing a barcode.
NB:WB = NS:WS =1:2.5 (Recommended ratio)

The width of a narrow bar is a key for selecting a barcode reader.
Narrow bar width is also named 'minimum element width'.

One Point

Binary Level and Multiple Level

Barcodes in the categories of CODE 39, CODABAR and ITF consist of bars and spaces that have two steps of narrow and wide widths. It is named the 'binary level' barcode.
The ratio between narrow and wide is 1 : 2 to 1 : 3. Some allowance is possible.

Barcodes in the categories of EAN and CODE 128 consist of bars and spaces that have four steps of narrow and wide widths. It is named the “multiple level” barcode.
The ratio is 1 : 2 : 3 : 4. Allowance is very limited.

One Point

'Multiple level' EAN and CODE 128 have four steps of bar width.
If the print quality is not good enough to identify the bar width, reading error tends to occur.
The printers with poor print quality including dot impact printer are not suitable for EAN and CODE 128.

What Is Check Digit?

A check digit is a numeric value calculated to check for read error.
The following explains the check flow and calculation method.

• The calculated check digit is compared with that appended to the read barcode.
• If they do not match, a read error occurs.

One Point

Right Qr 1 2 X 4

Right Qr 1 2