The Least Mean Squares Algorithm
Many of the examples come from graph theory, but the range is quite wide, from Fibonacci numbers to error-correcting codes with many stops in between. Most of the miniatures in the book require quite a bit of linear algebra, so they are not intended for use in a first linear algebra course. None of this is difficult, but it is not necessarily stuff that one would find in an introductory course.
For me, the biggest impact of the book came from noticing the tools that are used. Similarly, the theorem relating the rank of the product of two matrices to the ranks of the factors plays a big role here. Most linear algebra instructors would benefit from this kind of insight.
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Thirty-three Miniatures would be an excellent book for an informal seminar offered to students after their first linear algebra course. It may also be the germ of many interesting undergraduate talks. Skip to main content. Search form Search. Login Join Give Shops. Halmos - Lester R. Ford Awards Merten M.
Publication Date:. The identifier is the return value. This method can be used explicitly, e. This method can be used explicitly or implicitly. We will not do this in the following for typographical reasons, but note that it is often preferable in user code. Unfortunately, Octave version 4. To work around this issue, we provide a simple function import.
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The commands above would be equivalent to: 3. The syntax is also the same, except for functions such as SX. DM is mainly used for storing matrices in CasADi and as inputs and outputs of functions.
It is not intended to be used for computationally intensive calculations. Note how the multiplication and the addition were performed elementwise and new expressions of type SX were created for each entry of the result matrix. We shall now introduce a second, more general matrix expression type MX. The MX type allows, like SX , to build up expressions consisting of a sequence of elementary operations.
But unlike SX , these elementary operations are not restricted to be scalar unary or binary operations or. Instead, the elementary operations that are used to form MX expressions are allowed to be general multiple sparse-matrix valued input, multiple sparse-matrix valued output functions:. As a consequence, MX can be more economical when working with operations that are naturally vector or matrix valued with many elements. MX supports getting and setting elements, using the same syntax as SX , but the way it is implemented is very different.
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Element access and assignment, of the type you have just seen, are examples of operations that can be used to construct expressions. Other examples of operations are matrix multiplications, transposes, concatenations, resizings, reshapings and function calls. You can, however, in an MX graph include calls to a function defined by SX expressions.
Computational and Algorithmic Linear Algebra and n-Dimensional Geometry
Mixing SX and MX is often a good idea since functions defined by SX expressions have a much lower overhead per operation making it much faster for operations that are naturally written as a sequence of scalar operations. This is a standard format for sparse matrices that allows linear algebra operations such as elementwise operations, matrix multiplication and transposes to be performed efficiently.
In the CCS format, the sparsity pattern is decoded using the dimensions - the number of rows and number of columns - and two vectors. The first vector contains the index of the first structurally nonzero element of each column and the second vector contains the row index for every nonzero element. For more details on the CCS format, see e. Templates for the Solution of Linear Systems on Netlib. Sparsity patterns in CasADi are stored as instances of the Sparsity class, which is reference-counted , meaning that multiple matrices can share the same sparsity pattern, including MX expression graphs and instances of SX and DM.