Category Archives: what-is

What Is the Growth Factor for Gaussian Elimination?

Gaussian elimination is the process of reducing an matrix to upper triangular form by elementary row operations. It consists of stages, in the th of which multiples of row are added to later rows to eliminate elements below the diagonal … Continue reading

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What Is Stochastic Rounding?

In finite precision arithmetic the result of an elementary arithmetic operation does not generally lie in the underlying number system, , so it must be mapped back into by the process called rounding. The most common choice is round to … Continue reading

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What Is the Hilbert Matrix?

The Hilbert matrix is the matrix with . For example, It is probably the most famous test matrix and its conditioning and other properties were extensively studied in the early days of digital computing, especially by John Todd. The Hilbert … Continue reading

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What Is a Fréchet Derivative?

Let and be Banach spaces (complete normed vector spaces). The Fréchet derivative of a function at is a linear mapping such that for all . The notation should be read as “the Fréchet derivative of at in the direction ”. … Continue reading

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What Is the Adjugate of a Matrix?

The adjugate of an matrix is defined by where denotes the submatrix of obtained by deleting row and column . It is the transposed matrix of cofactors. The adjugate is sometimes called the (classical) adjoint and is sometimes written as … Continue reading

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What Is a Matrix Function?

If you give an array as the argument to a mathematical function in a programming language or problem solving environment you are likely to receive the result of applying the function to each component of the array. For example, here … Continue reading

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What Is Bfloat16 Arithmetic?

Bfloat16 is a floating-point number format proposed by Google. The name stands for “Brain Floating Point Format” and it originates from the Google Brain artificial intelligence research group at Google. Bfloat16 is a 16-bit, base 2 storage format that allocates … Continue reading

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What Is the Matrix Exponential?

The exponential of a square matrix is defined by the power series (introduced by Laguerre in 1867) That the series converges follows from the convergence of the series for scalars. Various other formulas are available, such as The matrix exponential … Continue reading

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What Is a Matrix Square Root?

A square root of an matrix is any matrix such that . For a scalar (), there are two square roots (which are equal if ), and they are real if and only if is real and nonnegative. For , … Continue reading

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What Is IEEE Standard Arithmetic?

The IEEE Standard 754, published in 1985 and revised in 2008 and 2019, is a standard for binary and decimal floating-point arithmetic. The standard for decimal arithmetic (IEEE Standard 854) was separate when it was first published in 1987, but … Continue reading

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