By Charles L Byrne
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Additional info for Applied iterative methods
1 Linear and Affine Linear Operators For example, if X = C J and A is a J by J complex matrix, then we can deﬁne an operator T by setting T x = Ax, for each x in C J ; here Ax denotes the multiplication of the matrix A and the column vector x. 20. An operator T is said to be a linear operator if T (αx + βy) = αT x + βT y, for each pair of vectors x and y and each pair of scalars α and β. Any operator T that comes from matrix multiplication, that is, for which T x = Ax, is linear. 21. For H = H(a, γ), H0 = H(a, 0), and any x and y in X , we have PH (x + y) = PH x + PH y − PH 0, so that PH0 (x + y) = PH0 x + PH0 y, that is, the operator PH0 is an additive operator.
The detected signals are related to the distribution of interest by means of the Fourier transform. Which values of the Fourier transform we obtain depend on the particular magnetic ﬁelds activated. In some approaches, the data are line integrals, as in tomography. Both iterative and noniterative methods can be used to obtain the reconstruction. 6 Intensity-Modulated Radiation Therapy The problem in intensity-modulated radiation therapy (IMRT) is to determine the various intensities of radiation to apply to the patient so as to deliver the desired minimum dosage to the tumor, while not exceeding the acceptable dosage to nearby parts of the body.
YJ )T in C J , the dot product x · y is deﬁned to be J x·y = xj yj . j=1 Note that we can write x · y = y † x. The norm, or Euclidean length, of x is √ √ ||x||2 = x · x = x† x. As in the real case, the distance between vectors x and y is ||x − y||2 . Both of the spaces RJ and C J , along with their dot products, are examples of ﬁnite-dimensional Hilbert space. Much of what follows in this chapter applies to both RJ and C J . In such cases, we shall simply refer to the underlying space as X . 1.
Applied iterative methods by Charles L Byrne