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In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers, then the complex conjugate of a + b i {\displaystyle a+bi} is a − b i . {\displaystyle a-bi.}
Powell's method, strictly Powell's conjugate direction method, is an algorithm proposed by Michael J. D. Powell for finding a local minimum of a function. The function need not be differentiable, and no derivatives are taken. The function must be a real-valued function of a fixed number of real-valued inputs. The caller passes in the initial point.
The letter stands for a vector in , is a complex number, and ¯ denotes the complex conjugate of . [1] More concretely, the complex conjugate vector space is the same underlying real vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate linear complex structure J {\displaystyle J} (different ...
The conjugate transpose of a matrix with real entries reduces to the transpose of , as the conjugate of a real number is the number itself. The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2 × 2 {\displaystyle 2\times 2} real matrices, obeying matrix addition and multiplication: a + i b ≡ ...
Conjugate gradient, assuming exact arithmetic, converges in at most n steps, where n is the size of the matrix of the system (here n = 2). In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-semidefinite.
It happens to be the unique maximal ideal of the ring of dual numbers. The group of units of the dual number ring then consists of numbers not in the ideal. The dual numbers form a local ring since there is a unique maximal ideal. The group of units is a Lie group and can be studied using the exponential mapping.
If = is self-adjoint, = and =, then =, =, and the conjugate gradient method produces the same sequence = at half the computational cost.; The sequences produced by the algorithm are biorthogonal, i.e., = = for .
The other prime numbers are not Gaussian primes, but each is the product of two conjugate Gaussian primes. A Gaussian integer a + bi is a Gaussian prime if and only if either: one of a, b is zero and the absolute value of the other is a prime number of the form 4n + 3 (with n a nonnegative integer), or