ZGOUBI USERS' GUIDE - HEP

in fourth order. The coefficients ´The ´^·^^^¡ >¡ MM¡>´ ´ §§qTP´7´ ´ M´´>§§T>T´q@§T´7 §>§P>>§>>>´§P @ \ ´´ >´ T§MT 47 Tare calculated by expressions that minimize, with respect to ´· fmay then be identified with the derivatives of ¡ ^^¡¡´q^‰bb ´g´§>>´4 >TT´T§M4 > ´P4 >>4 P>1.4 Calculation of ¡ from Field Maps 21 @ R N§#" \ ´§ U§#" \ @(1.4.3)^ §> G§ ^ §B B B1.4.2 2-D Median Plane Map, with Median Plane AntisymmetryLet be the value of ¡|• §©"$Œ" \ at the nodes of a mesh which defines a 2-D field map in the (§#"$ plane while¡ Ž ¡ \ bhg and ¡|– §#"kS" \ are assumed to be zero. Such a map may have been built or measured in either Cartesian§#"kS" or polar coordinates. Whenever polar coordinates are used, a change to Cartesian coordinates (described below) providesthe expression of and its derivatives as involved in eq. (1.2.8).zgoubi provides three types of polynomial interpolation from the mesh (option IORDRE); namely, a second order interpolation,with either a 9- or a 25-point grid, or a fourth order interpolation with a 25-point grid (Fig. ¡ 3).If the 2-D field map is built up from simulation, the grid simply aims at interpolating the field at a given point from its 9or 25 neighbors. If the map results from measurements, the grid also smoothes field measurement fluctuations.The mesh may be defined in Cartesian coordinates, (Figs. 3A and 3B) or in polar coordinates (Fig. 3C).The interpolation grid is centered on the node which is closest to the projection (§#"k in the plane of the actual point ofthe trajectory.The interpolation polynomial is§#" \ @ \ ´^ §in second order, or§©"$Œ" \ ´¡ ´§ ´ ´> 4 ´§© ´(1.4.4)4k47ˆ44‡77k77$7¡ 4k4§#"kS" \ ´7ˆ44‡7§ ´ ´> 4´ 4 T> 7 ´P(1.4.5)§© ´ ´§©, the quadratic sumP 4T 7 ´>$>§©big(1.4.6)bhg ¡ §©"$Œ" \ w6¡ bhg The source code contains the explicit analytical expressions of the ´bigcoefficients.†\solutions of the normal equationsbhgbig^ ´bhg§©"$Œ" \ at the central node of the gridg ¡\ " \ " \ (1.4.7)bhg and , at the actual pointinterpolation polynomial, which gives (e.g. from (1.4.4) in the case of second order interpolation)The derivatives §#"$Œ" \ of with respect § to ¡^ §^ §#"kS" \ are obtained by differentiation of theA š A7¸47$7^ §(1.4.8)§ ´§#"$Œ" \ ´ @> 47k7^ etc.4e7§#"$Œ" \ ´§ @This allows stepping to the calculation §#"kS"e of and its derivatives as described in subsection 1.3.2 (eq. 1.3.3).¡The special case of polar mapsIt is necessary to change from polar map frame ( 1 "$¹Œ"[ ) to the Cartesian moving frame (§#"$Œ"[ ). This is done as follows.