f SVD Singular Value Decomposition Alexander Valishev Accelerator

f SVD Singular Value Decomposition Сингулярное разложение матриц Alexander Valishev Accelerator Summer Students Meeting August 13, 2007

f History § First introduced by Eugenio Beltrami 1873, Camille Jordan 1874 § Proof - Eckart, Young 1936 § Emile Picard in 1910 introduced the term ‘singluar value’ § Practical computing method – Golub, Kahan 1965 Accelerator Summer Students Mtg. 8/13/07 – A. Valishev 2

f Definition M is any mxn matrix. U is mxm, V nxn Accelerator Summer Students Mtg. 8/13/07 – A. Valishev 3

f Definition Singular value s u – left-singular, v - right-singular vectors If si are not degenerate and ≠ 0 – U and V are unique Accelerator Summer Students Mtg. 8/13/07 – A. Valishev 4

f Pseudo-Inverse Matrix 1/si if si ≠ 0 0 si = 0 Practical application: 1/si if si > e, 0 otherwise Accelerator Summer Students Mtg. 8/13/07 – A. Valishev 5

f Example 1 Accelerator Summer Students Mtg. 8/13/07 – A. Valishev #1 6

f Example 1 #2 M x = y -> x = M-1 y |M x - y| = min Accelerator Summer Students Mtg. 8/13/07 – A. Valishev 7

f Example 2 Accelerator Summer Students Mtg. 8/13/07 – A. Valishev #1 8

f Example 2 #2 M x = y -> x = M-1 y Accelerator Summer Students Mtg. 8/13/07 – A. Valishev 9

f Example 2 #3 |M x 2 - y| = min Accelerator Summer Students Mtg. 8/13/07 – A. Valishev 10

f Example 3 Accelerator Summer Students Mtg. 8/13/07 – A. Valishev #1 11

f Example 3 #2 M x = y -> x = M-1 y |x| = min Accelerator Summer Students Mtg. 8/13/07 – A. Valishev 12

f Differential Orbit Measurements § The aim is to find gradient errors utilizing the fact that quadrupoles act as dipole correctors with off-center orbit § Initially, closed orbit is excited using a single dipole corrector § The orbit distortion due to quadrupoles is given by § Dispersion measurement § Use BPM system to measure and record orbit differences Accelerator Summer Students Mtg. 8/13/07 – A. Valishev 13

f Orbit response matrix fit § The orbit response matrix is the change in the orbit at the BPMs as a function of changes in steering magnets: (LOCO, V. Sajaev, ANL) • Modern storage rings have a large number of steering magnets and precise BPMs, so measurement of the response matrix provides a very large array of precisely measured data • The response matrix is defined by the linear lattice of the machine; therefore it can be used to calibrate the linear optics in a storage ring Accelerator Summer Students Mtg. 8/13/07 – A. Valishev 14

f Orbit response matrix fit The response matrix depends on the following parameters: § § § § § Quadrupole gradient errors Steering magnet calibrations BPM gains Quadrupole tilts Steering magnet tilts BPM tilts Energy shift associated with steering magnet changes BPM nonlinearity Steering magnet and BPM longitudinal positions etc. Accelerator Summer Students Mtg. 8/13/07 – A. Valishev Main parameters Main coupling parameters 15

f Orbit response matrix fit for Tevatron § Tevatron has 110 steering magnets and 120 BPMs in each plane and 216 quadrupoles For our analysis we use about 30 steering magnets in each plane, all BPMs, all quadrupoles, and tilts of one half of quadrupoles. The resulting response matrix has about 16, 500 elements, and the number of variables is 980. Finally we solve the following equation (by iterations): § § X = M-1 · V 130 Mb Accelerator Summer Students Mtg. 8/13/07 – A. Valishev 16