Successive Approximations: Sheet 2

SHEET 2A. Excel functions that compute COMPLEX solutions of w⁹ - 23w² + 5w - 19 = 0

A B C D E F G H
1 X Y R THETA Fr Fi Sr Si 1
2 = 1 = 1 = SQRT(A2^2 + B2^2) = IF(B2<0, -ACOS(A2/C2), +ACOS(A2/C2)) = C2^9*COS(9*D2) - 23*C2^2*COS(2*D2) + 5*C2*COS(D2) - 19 = C2^9*SIN(9*D2) - 23*C2^2*SIN(2*D2) + 5*C2*SIN(D2) = 9*C2^8*COS(8*D2) - 46*C2*COS(D2) + 5 = 9*C2^8*SIN(8*D2) - 46*C2*SIN(D2) 2
3 = A2 - (E2*G2 + F2*H2)/(G2^2 + H2^2) = B2 - (F2*G2 - E2*H2)/(G2^2 + H2^2) = SQRT(A3^2 + B3^2) = IF(B3<0, -ACOS(A3/C3), +ACOS(A3/C3)) = C3^9*COS(9*D3) - 23*C3^2*COS(2*D3) + 5*C3*COS(D3) - 19 = C3^9*SIN(9*D3) - 23*C3^2*SIN(2*D3) + 5*C3*SIN(D3) = 9*C3^8*COS(8*D3) - 46*C3*COS(D3) + 5 = 9*C3^8*SIN(8*D3) - 46*C3*SIN(D3) 3
4 = A3 - (E3*G3 + F3*H3)/(G3^2 + H3^2) = B3 - (F3*G3 - E3*H3)/(G3^2 + H3^2) = SQRT(A4^2 + B4^2) = IF(B4<0, -ACOS(A4/C4), +ACOS(A4/C4)) = C4^9*COS(9*D4) - 23*C4^2*COS(2*D4) + 5*C4*COS(D4) - 19 = C4^9*SIN(9*D4) - 23*C4^2*SIN(2*D4) + 5*C4*SIN(D4) = 9*C4^8*COS(8*D4) - 46*C4*COS(D4) + 5 = 9*C4^8*SIN(8*D4) - 46*C4*SIN(D4) 4

SHEET 2B. Values returned by functions in SHEET 2A.

A B C D E F G H
1 X Y R THETA Fr Fi Sr Si 1
2 1 1 1.41421 0.78540 2 -25 103 -46 2
3 0.89343811 1.19512770 1.49217 0.92885 -17.8063 -11.05841 54.73 146.71 3
4 0.99935027 1.11327154 1.49602 0.83927 2.64429 -9.75850 164.19 43.12 4
5 0.99888690 1.17282732 1.54055 0.86532 -2.08949 0.74234 188.18 116.43 5
6 1.00515181 1.16500635 1.53869 0.85893 -0.03251 -0.06423 194.02 103.32 6
7 1.00541969 1.16519475 1.53901 0.85887 4.47E-05 6.208E-05 194.46 103.47 7
8 1.00541938 1.16519460 1.53901 0.85887 6.94E-11 5.111E-11 194.46 103.47 8
9 1.00541938 1.16519460 1.53901 0.85887 0 -9.5E-14 194.46 103.47 9
A B C D E F G H

URL http://members.ispwest.com/patrussell/approximations/Sheet_2.htm
Unpublished Work. © Copyright 2001 Pat Russell. Updated April 27, 2003.