By (Eds.) G. Senjanovic, A. Smirnov, G. Thompson

ISBN-10: 9295003020

ISBN-13: 9789295003026

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**Example text**

36) says that the residual vector r is perpendicular to each row in the F matrix. These rows are the fitting functions. Therefore, the residual, after it has been minimized, is perpendicular to all the fitting functions. 3 Differentiation by a complex vector Complex numbers frequently arise in physical problems, particularly those with Fourier series. 2. MULTIVARIATE LEAST SQUARES 43 x. 5). Use a prime, such as x , to denote the complex conjugate of the transposed vector x. 4), we minimized a quadratic form Q( X¯ , X ) by setting to zero both ∂ Q/∂ X¯ and ∂ Q/∂ X .

Mathematicians define adjoints by this test, except that instead of using random vectors, they say “for all functions,” which includes the continuum. This defining test makes adjoints look mysterious. Careful inspection of operator adjoints, however, generally reveals that they are built up from simple matrices. Given adjoints A , B , and C , the adjoint of ABC is C B A . Fourier transforms and linear-differential-equation solvers are chains of matrices, so their adjoints can be assembled by the application of adjoint components in reverse order.

What we find in practice is that an application and its adjoint amounts to two routines. The first routine amounts to the matrix multiplication Fx. The adjoint routine computes F y, where F is the conjugate-transpose matrix. In later chapters we will be solving huge sets of simultaneous equations, in which both routines are required. If the pair of routines are inconsistent, we are doomed from the start. The dot-product test is a simple test for verifying that the two routines are adjoint to each other.

### Astroparticle Physics and Cosmology by (Eds.) G. Senjanovic, A. Smirnov, G. Thompson

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