For large matrices, special attention is paid to cache-friendliness.

Fixed-size matrices are fully optimized: dynamic memory allocation is avoided, and the loops are unrolled when that makes sense.

Explicit vectorization is performed for SSE 2/3/4, AVX, AVX2, FMA, AVX512, ARM NEON (32-bit and 64-bit), PowerPC AltiVec/VSX (32-bit and 64-bit), ZVector (s390x/zEC13) SIMD instruction sets, and since 3.4 MIPS MSA with graceful fallback to non-vectorized code.

Expression templates allow intelligently removing temporaries and enable lazy evaluation, when that is appropriate.

Its ecosystem of unsupported modules provides many specialized features such as non-linear optimization, matrix functions, a polynomial solver, FFT, and much more.

It supports various matrix decompositions and geometry features.

It supports all standard numeric types, including std::complex, integers, and is easily extensible to custom numeric types.

It supports all matrix sizes, from small fixed-size matrices to arbitrarily large dense matrices, and even sparse matrices.

Lapack_lug.html), Third Edition, SIAM, Philadelphia, 1999. Inv uses LAPACK routines to compute the matrix inverse: Using A instead of inv(A)*b is two to three times as fast and produces residuals on the order of machine accuracy, relative to the magnitude of the data. The direct solution produces residuals on the order of the machine accuracy, even though the system is badly conditioned. But the size of the residuals, obtained by plugging the computed solution back into the original equations, differs by several orders of magnitude. Both produce computed solutions with about the same error, 1.e-6, reflecting the condition number of the matrix. It takes almost two and one half times as long to compute the solution with y = inv(A)*b as with z = A. On a 300 MHz, laptop computer the statements Thus the system of linear equations is badly conditioned, but consistent. The exact solution x is a random vector of length 500 and the right-hand side is b = A*x. A random matrix A of order 500 is constructed so that its condition number, cond(A), is 1.e10, and its norm, norm(A), is 1. Here is an example demonstrating the difference between solving a linear system by inverting the matrix with inv(A)*b and solving it directly with A. This produces the solution using Gaussian elimination, without forming the inverse. A better way, from both an execution time and numerical accuracy standpoint, is to use the matrix division operator x = A.

One way to solve this is with x = inv(A)*b. A frequent misuse of inv arises when solving the system of linear equation s. In practice, it is seldom necessary to form the explicit inverse of a matrix. A warning message is printed if X is badly scaled or nearly singular. Returns the inverse of the square matrix X. Inv (MATLAB Functions) MATLAB Function Reference

Matrix inverse matlab