what-is

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• What-Is

This is a series of ``What Is" articles, which give brief descriptions of important concepts in numerical analysis and related areas, with a focus on topics that arise in my research.

My aim is for the articles to adhere to the following criteria.

• The posts are no more than two or three screenfuls (about 500 words).
• The posts contain a minimum of mathematical symbols and equations and a minimum of citations.
• They are accessible to upper level undergraduate students in any mathematically oriented subject.
• The list of references given for further reading is very small and concentrates on those that are broad, readable, and contain further references to the literature.

The articles are available in blog form at https://nhigham.com/category/what-is/ and are provided in PDF form in this repository. The PDF files contain hyperlinks, which are clickable in the PDF but not visible in a printout.

The articles are (in order of publication)

• [[https://github.com/higham/what-is/blob/master/cond.pdf][What Is a Condition Number?]]
• [[https://github.com/higham/what-is/blob/master/berr.pdf][What Is Backward Error?]]
• [[https://github.com/higham/what-is/blob/master/matrix.pdf][What Is a Matrix?]]
• [[https://github.com/higham/what-is/blob/master/geninv.pdf][What Is a Generalized Inverse?]]
• [[https://github.com/higham/what-is/blob/master/orthog.pdf][What Is an Orthogonal Matrix?]]
• [[https://github.com/higham/what-is/blob/master/hadamard.pdf][What Is a Hadamard Matrix?]]
• [[https://github.com/higham/what-is/blob/master/correlation.pdf][What Is a Correlation Matrix?]]
• [[https://github.com/higham/what-is/blob/master/randorth.pdf][What Is a Random Orthogonal Matrix?]]
• [[https://github.com/higham/what-is/blob/master/rounding.pdf][What Is Rounding?]]
• [[https://github.com/higham/what-is/blob/master/float_arith.pdf][What Is Floating-Point Arithmetic?]]
• [[https://github.com/higham/what-is/blob/master/ieee_arith.pdf][What Is IEEE Standard Arithmetic?]]
• [[https://github.com/higham/what-is/blob/master/square_root.pdf][What Is a Matrix Square Root?]]
• [[https://github.com/higham/what-is/blob/master/matrix_exponential.pdf][What Is the Matrix Exponential?]]
• [[https://github.com/higham/what-is/blob/master/bfloat16.pdf][What Is Bfloat16 Arithmetic?]]
• [[https://github.com/higham/what-is/blob/master/matrix_function.pdf][What Is a Matrix Function?]]
• [[https://github.com/higham/what-is/blob/master/adjugate.pdf][What Is the Adjugate of a Matrix?]]
• [[https://github.com/higham/what-is/blob/master/frechet.pdf][What Is a Fréchet Derivative?]]
• [[https://github.com/higham/what-is/blob/master/hilbert_matrix.pdf][What Is the Hilbert Matrix?]]
• [[https://github.com/higham/what-is/blob/master/stochround.pdf][What Is Stochastic Rounding?]]
• [[https://github.com/higham/what-is/blob/master/growth_factor.pdf][What Is the Growth Factor for Gaussian Elimination?]]
• [[https://github.com/higham/what-is/blob/master/symm_pos_def.pdf][What Is a Symmetric Positive Definite Matrix?]]
• [[https://github.com/higham/what-is/blob/master/polar_decomp.pdf][What Is the Polar Decomposition?]]
• [[https://github.com/higham/what-is/blob/master/numerical-stability.pdf][What Is Numerical Stability?]]
• [[https://github.com/higham/what-is/blob/master/cholesky.pdf][What Is a Cholesky Factorization?]]
• [[https://github.com/higham/what-is/blob/master/gerstenhaber.pdf][What Is the Gerstenhaber Problem?]]
• [[https://github.com/higham/what-is/blob/master/kronprod.pdf][What Is the Kronecker Probduct?]]
• [[https://github.com/higham/what-is/blob/master/sylvester_eqn.pdf][What Is the Sylvester Equation?]]
• [[https://github.com/higham/what-is/blob/master/sparse.pdf][What Is a Sparse Matrix?]]
• [[https://github.com/higham/what-is/blob/master/householder.pdf][What Is a Housholder Matrix?]]
• [[https://github.com/higham/what-is/blob/master/block_matrix.pdf][What Is a Block Matrix?]]
• [[https://github.com/higham/what-is/blob/master/sherman_morrison.pdf][What Is the Sherman--Morrison--Woodbury Formula?]]
• [[https://github.com/higham/what-is/blob/master/complex_step.pdf][What Is the Complex Step Approximation?]]
• [[https://github.com/higham/what-is/blob/master/svd.pdf][What Is the Singular Value Decomposition?]]
• [[https://github.com/higham/what-is/blob/master/csd.pdf][What Is the CS Decomposition?]]
• [[https://github.com/higham/what-is/blob/master/cayley-hamilton.pdf][What Is the Cayley--Hamilton Theorem?]]
• [[https://github.com/higham/what-is/blob/master/qr.pdf][What Is a QR Factorization?]]
• [[https://github.com/higham/what-is/blob/master/matrix_logarithm.pdf][What Is the Matrix Logarithm?]]
• [[https://github.com/higham/what-is/blob/master/normal.pdf][What Is a (Non)normal Matrix?]]
• [[https://github.com/higham/what-is/blob/master/unwinding.pdf][What Is the Matrix Unwinding Function?]]
• [[https://github.com/higham/what-is/blob/master/fractional_power.pdf][What Is a Fractional Matrix Power?]]
• [[https://github.com/higham/what-is/blob/master/matrix_sign.pdf][What Is the Matrix Sign Function?]]
• [[https://github.com/higham/what-is/blob/master/mod_chol.pdf][What Is a Modified Cholesky Factorization?]]
• [[https://github.com/higham/what-is/blob/master/logsumexp.pdf][What Is the Log-Sum-Exp Function?]]
• [[https://github.com/higham/what-is/blob/master/softmax.pdf][What Is the Softmax Function?]]
• [[https://github.com/higham/what-is/blob/master/near_symm.pdf][What Is the Nearest Symmetric Matrix?]]
• [[https://github.com/higham/what-is/blob/master/near_psd.pdf][What Is the Nearest Positive Semidefinite Matrix?]]
• [[https://github.com/higham/what-is/blob/master/unit_inv_norm.pdf][What Is a Unitarily Invariant Norm?]]
• [[https://github.com/higham/what-is/blob/master/m-matrix.pdf][What Is an M-Matrix?]]
• [[https://github.com/higham/what-is/blob/master/companion.pdf][What Is a Companion Matrix?]]
• [[https://github.com/higham/what-is/blob/master/diag_dom.pdf][What Is a Diagonally Dominant Matrix?]]
• [[https://github.com/higham/what-is/blob/master/lu.pdf][What Is an LU Factorization?]]
• [[https://github.com/higham/what-is/blob/master/pseudo_orthog.pdf][What Is a Pseudo-Orthogonal Matrix?]]
• [[https://github.com/higham/what-is/blob/master/rrf.pdf][What Is a Rank-Revealing Factorization?]]
• [[https://github.com/higham/what-is/blob/master/wilson.pdf][What Is the Wilson Matrix?]]
• [[https://github.com/higham/what-is/blob/master/vandermonde.pdf][What Is a Vandermonde Matrix?]]
• [[https://github.com/higham/what-is/blob/master/determinant.pdf][What Is the Determinant of a Matrix?]]
• [[https://github.com/higham/what-is/blob/master/kms.pdf][What Is the Kac–Murdock-Szegö Matrix?]]
• [[https://github.com/higham/what-is/blob/master/perron-frobenius.pdf][What Is the Perron--Frobenius Theorem?]]
• [[https://github.com/higham/what-is/blob/master/totally_nonneg.pdf][What Is a Totally Nonnegative Matrix?]]
• [[https://github.com/higham/what-is/blob/master/norm_vector.pdf][What Is a Vector Norm?]]
• [[https://github.com/higham/what-is/blob/master/norm_matrix.pdf][What Is a Matrix Norm?]]
• [[https://github.com/higham/what-is/blob/master/blocked.pdf][What Is a Blocked Algorithm?]]
• [[https://github.com/higham/what-is/blob/master/tridiag.pdf][What Is a Tridiagonal Matrix?]]
• [[https://github.com/higham/what-is/blob/master/log_norm.pdf][What Is the Logarithmic Norm?]]
• [[https://github.com/higham/what-is/blob/master/frank.pdf][What Is the Frank Matrix?]]
• [[https://github.com/higham/what-is/blob/master/second_diff.pdf][What Is the Second Difference Matrix?]]
• [[https://github.com/higham/what-is/blob/master/jcf.pdf][What Is the Jordan Canonical Form?]]
• [[https://github.com/higham/what-is/blob/master/backslash.pdf][What Is A\A?]]
• [[https://github.com/higham/what-is/blob/master/inverse.pdf][What Is the Matrix Inverse?]]
• [[https://github.com/higham/what-is/blob/master/permutation.pdf][What Is a Permutation Matrix?]]
• [[https://github.com/higham/what-is/blob/master/schur.pdf][What Is a Schur Decomposition?]]
• [[https://github.com/higham/what-is/blob/master/pascal.pdf][What Is the Pascal Matrix?]]