Florette Dikkers

The pivot or pivot element is the element of a matrix, or an array, which is selected first by an algorithm (e.g. Gaussian elimination, simplex algorithm, etc.), to do certain calculations. In the case of matrix algorithms, a pivot entry is usually required to be at least distinct from zero, and often distant from it; in this case finding this element is called pivoting. Pivoting may be followed by an interchange of rows or columns to bring the pivot to a fixed position and allow the algorithm to proceed successfully, and possibly to reduce round-off error. It is often used for verifying row echelon form.

Pivoting might be thought of as swapping or sorting rows or columns in a matrix, and thus it can be represented as multiplication by permutation matrices. However, algorithms rarely move the matrix elements because this would cost too much time; instead, they just keep track of the permutations.

Overall, pivoting adds more operations to the computational cost of an algorithm. These additional operations are sometimes necessary for the algorithm to work at all. Other times these additional operations are worthwhile because they add numerical stability to the final result.

In the case of Gaussian elimination, the algorithm requires that pivot elements not be zero. Interchanging rows or columns in the case of a zero pivot element is necessary. The system below requires the interchange of rows 2 and 3 to perform elimination.

The system that results from pivoting is as follows and will allow the elimination algorithm and backwards substitution to output the solution to the system.

Furthermore, in Gaussian elimination it is generally desirable to choose a pivot element with large absolute value. This improves the numerical stability. The following system is dramatically affected by round-off error when Gaussian elimination and backwards substitution are performed.

This system has the exact solution of x1 = 10.00 and x2 = 1.000, but when the elimination algorithm and backwards substitution are performed using four-digit arithmetic, the small value of a11 causes small round-off errors to be propagated. The algorithm without pivoting yields the approximation of x1 ≈ 9873.3 and x2 ≈ 4. In this case it is desirable that we interchange the two rows so that a21 is in the pivot position

Considering this system, the elimination algorithm and backwards substitution using four-digit arithmetic yield the correct values x1 = 10.00 and x2 = 1.000.

In partial pivoting, the algorithm selects the entry with largest absolute value from the column of the matrix that is currently being considered as the pivot element. More specifically, when reducing a matrix to row echelon form, partial pivoting swaps rows before the column's row reduction to make the pivot element have the largest absolute value compared to the elements below in the same column. Partial pivoting is generally sufficient to adequately reduce round-off error.

However, for certain systems and algorithms, complete pivoting (or maximal pivoting) may be required for acceptable accuracy. Complete pivoting interchanges both rows and columns in order to use the largest (by absolute value) element in the matrix as the pivot. Complete pivoting is usually not necessary to ensure numerical stability and, due to the additional cost of searching for the maximal element, the improvement in numerical stability that it provides is typically outweighed by its reduced efficiency for all but the smallest matrices. Hence, it is rarely used.

Another strategy, known as rook pivoting also interchanges both rows and columns but only guarantees that the chosen pivot is simultaneously the largest possible entry in its row and the largest possible entry in its column, as opposed to the largest possible in the entire remaining submatrix. When implemented on serial computers this strategy has expected cost of only about three times that of partial pivoting and is therefore cheaper than complete pivoting. Rook pivoting has been proved to be more stable than partial pivoting both theoretically and in practice.

A variation of the partial pivoting strategy is scaled pivoting. In this approach, the algorithm selects as the pivot element the entry that is largest relative to the entries in its row. This strategy is desirable when entries' large differences in magnitude lead to the propagation of round-off error. Scaled pivoting should be used in a system like the one below where a row's entries vary greatly in magnitude. In the example below, it would be desirable to interchange the two rows because the current pivot element 30 is larger than 5.291 but it is relatively small compared with the other entries in its row. Without row interchange in this case, rounding errors will be propagated as in the previous example.

A pivot position in a matrix, A, is a position in the matrix that corresponds to a row–leading 1 in the reduced row echelon form of A. Since the reduced row echelon form of A is unique, the pivot positions are uniquely determined and do not depend on whether or not row interchanges are performed in the reduction process. Also, the pivot of a row must appear to the right of the pivot in the above row in row echelon form.

This article incorporates material from Pivoting on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

What is VAMT - Microsoft Volume Activation Management Toolkit is a free tool to manage and apply Microsoft licenses over the network. VAMT can manage volume activation using Multiple Activation Keys (MAKs) or the Windows Key Management Service (KMS). The tool can be loaded with your volume licensing keys and when required each machine can be queried for its license state and activate from the preloaded keys. Activations can be proxied through the VAMT installed machine, which means end-user machines does not require an internet connection.

When comparing with the previous version, latest VAMT requires you to download and install SQL database as a prerequisite, if you have SQL server you can use it or you can use the SQL Express version which is free to use. Here I’m downloading and installing SQL express 2017.

VAMT can be used with MAK keys and KMS keys to activate Windows servers, clients and Office products. The tool can be used to activate products using Online activation and Proxy activation.

Online Activation - Online activation enables you to activate over the Internet any products installed with MAK, KMS host, or retail product keys on one or more connected computers within the network. This process requires that each product communicate activation information directly to Microsoft. Each client should have internet access.

Proxy Activation – VAMT host distribute the keys MAK, KMS Host key (CSVLK), or retail product key to one or more client computers and generate a product ID, then this product ID is sent on behalf of the client to the Microsoft Activation Center. When the key is validated, confirmation ID is sent to the VAMT host. Using this confirmation, VAMT will activate its client computers. The main benefit of this is, clients does not have to connect to the internet to get it activated. Also, you can do multiple activations centrally managed.

In this scenario, I’m adding a Windows 2016 Datacenter MAK key. KMS host key also can be added using the same method.

Right-click on the Product Keys and select Add Product key. Type the product key and click Add Key.

When the product key is validated, it will successfully be added to the key list.

Before activation, you need to discover the machines which need the activation. This can be done through discovering products using Active directory search, workgroup search or manually entering the name or IP address.

If you are using KMS, you know there is not much to do when managing the KMS activations. However, if you want to control the KMS activations, see the activation count and manually activate clients using KMS key, you can install the VAMT on top of the KMS host. KMS can benefit from VAMT in the following key points:

Hope this post is useful

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A Guardian ad Litem must successfully complete 30 hours of certification training and 12 hours annually of re certification training , and spend an average of 10 hours per month working on the case (as with any average, some cases will involve more time, and some less time) What is the role of the Guardian ad Litem?

Baileys Irish Cream is an Irish cream liqueur ; an alcoholic drink flavoured with cream, cocoa and Irish whiskey It is made by Diageo at Nangor Road,

There are so many things to love about cask-wine finished whiskies. Here, we breakdown what they are, how they are produced, and look at some of our favourites.

Whisky producers age whisky in all sorts of casks - so much affects the taste of the whisky, from the wood the cask is made from, to the placement of the cask in the warehouse (the casks at the top of the warehouse actually taste different to the ones at the bottom!).

When wine is held in casks, even after it is removed from the cask, a small amount of wine soaks into the wood - so when that cask is refilled with whisky, the aroma and flavour of the wine seeps into the whisky over time. The whisky is usually aged for years before moving to the wine cask, so it is therefore 'finished' in the wine cask.

While you'd think there was some complex formula behind this, the real answer is - trial and error. Producers will add a few drops of wine to whisky and see how the flavours blend. Or - if they have a favourite wine and a favourite whisky, they can finish the whisky in that wine cask to combine their two favourite flavours.

The main reason: they are delicious! Also, it's a great way to experiment with flavour in a really subtle way - scotch, as a product is pretty heavily regulated, so finishing scotch in a wine cask, is a great way to create new flavours without totally altering the original product. A lot of whisky distillers also own wineries, reusing barrels is a great way to save resources + money.

Eirigh Na Greine (pronounced 'Ae-ree ne Gray-nyuh) means 'Morning Sky' in Galeic, a perfect name for this gorgeous amber gold drop. This very limited edition release comes from Bunnahabhain (proncounced 'Boon-ah-van'), from Islay in Scotland. It's matured in their warehouses before being finished in Italian and French red wine casks.

This whisky has a fruity, spicy aroma, with a rich, sweet palate and a smooth, lingering finish.

After maturing in traditional oak casks, this Single Malt was finished in Sauternes Wine Casks. Sauternes is a beloved sweet white wine from the Bordeaux region in France.

This wine is perfect for October - with notes of honey, apricot and pear, as well as fig and barley. A rich, sweet whisky perfect for warm spring afternoons.