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Author Archives: Nick Higham
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
Simulating Low Precision FloatingPoint Arithmetics in MATLAB
At least five floatingpoint arithmetics are available in mainstream hardware: the IEEE double precision (fp64), single precision (fp32), and half precision (fp16) formats, bfloat16, and tf32, introduced in the recently announced NVIDIA A100, which uses the NVIDIA Ampere GPU architecture. … Continue reading
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
1984 Symposium in Honour of James Wilkinson
In September 1984 a twoday Symposium on Computational Mathematics–State of the Art was held at Argonne National Laboratory in honour of James Hardy Wilkinson on his 65th birthday. Wilkinson was one of the leading figures in 20th century numerical analysis. … Continue reading
Singular Values of Rank1 Perturbations of an Orthogonal Matrix
What effect does a rank1 perturbation of norm 1 to an orthogonal matrix have on the extremal singular values of the matrix? Here, and throughout this post, the norm is the 2norm. The largest singular value of the perturbed matrix … Continue reading
Which Fountain Pen Ink for Writing Mathematics?
If, like me, you sometimes prefer to write mathematics on paper before typing into LaTeX, you have the choice of pencil or pen as your writing tool. I’ve written before about writing in pencil. My current tools of choice are … 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 floatingpoint arithmetic. The standard for decimal arithmetic (IEEE Standard 854) was separate when it was first published in 1987, but … Continue reading
What Is FloatingPoint Arithmetic?
A floatingpoint number system is a finite subset of the real line comprising numbers of the form where is the base, is the precision, and is the exponent. The system is completely defined by the four integers , , , … Continue reading
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What Is Rounding?
Rounding is the transformation of a number expressed in a particular base to a number with fewer digits. For example, in base 10 we might round the number to , which can be described as rounding to three significant digits … Continue reading
What Is a Random Orthogonal Matrix?
Various explicit parametrized formulas are available for constructing orthogonal matrices. To construct a random orthogonal matrix we can take such a formula and assign random values to the parameters. For example, a Householder matrix is orthogonal and symmetric and we … Continue reading
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