Index

SHEET 4A. Excel functions that compute COMPLEX solutions of w13w25 - 6w1w2 - 17 = 0 and 7w12w2 - 3w1w23 + 5 = 0
  A B C D E F G H I J K L M N O P Q R S T U V  
1 vector X vector Y R THETA vector Fr vector Fi matrix Jr matrix Ji matrix 1/Jr matrix 1/Ji matrix Marie = 1/Ji*Jr + 1/Jr*Ji matrix 1/Marie vector Vince = 1/Ji*Fr + 1/Jr*Fi vector Vito = 1/Ji*Fi - 1/Jr*Fr vector DX = 1/Marie * Vince vector DY = 1/Marie * Vito 1
2 1 -2 =SQRT(A2^2 + B2^2) =IF(B2<0, -ACOS(A2/C2), ACOS(A2/C2)) = C2^3*C3^5*COS(3*D2 + 5*D3) - 6*C2*C3*COS(D2 + D3) - 17 = C2^3*C3^5*SIN(3*D2 + 5*D3) - 6*C2*C3*SIN(D2 + D3) = 3*C2^2*C3^5*COS(2*D2 + 5*D3) - 6*C3*COS(D3) = 5*C2^3*C3^4*COS(3*D2 + 4*D3) - 6*C2*COS(D2) = 3*C2^2*C3^5*SIN(2*D2 + 5*D3) - 6*C3*SIN(D3) = 5*C2^3*C3^4*SIN(3*D2 + 4*D3) - 6*C2*SIN(D2) =INDEX(MINVERSE(G2:H3), 1, 1) =INDEX(MINVERSE(G2:H3), 1, 2) =INDEX(MINVERSE(I2:J3), 1, 1) =INDEX(MINVERSE(I2:J3), 1, 2) =INDEX(MMULT(M2:N3, G2:H3), 1, 1) + INDEX(MMULT(K2:L3, I2:J3), 1, 1) =INDEX(MMULT(M2:N3, G2:H3), 1, 2) + INDEX(MMULT(K2:L3, I2:J3), 1, 2) =INDEX(MINVERSE(O2:P3), 1, 1) =INDEX(MINVERSE(O2:P3), 1, 2) =INDEX(MMULT(M2:N3, E2:E3), 1, 1) + INDEX(MMULT(K2:L3, F2:F3), 1, 1) =INDEX(MMULT(M2:N3, F2:F3), 1, 1) - INDEX(MMULT(K2:L3, E2:E3), 1, 1) =INDEX(MMULT(Q2:R3, S2:S3), 1, 1) =INDEX(MMULT(Q2:R3, T2:T3), 1, 1) 2
3 1 -1 =SQRT(A3^2 + B3^2) =IF(B3<0, -ACOS(A3/C3), ACOS(A3/C3)) = 7*C2^2*C3*COS(2*D2 + D3) - 3*C2*C3^3*COS(D2 + 3*D3) + 5 = 7*C2^2*C3*SIN(2*D2 + D3) - 3*C2*C3^3*SIN(D2 + 3*D3) = 14*C2*C3*COS(D2 + D3) - 3*C3^3*COS(3*D3) = 7*C2^2*COS(2*D2) - 9*C2*C3^2*COS(D2 + 2*D3) = 14*C2*C3*SIN(D2 + D3) - 3*C3^3*SIN(3*D3) = 7*C2^2*SIN(2*D2) - 9*C2*C3^2*SIN(D2 + 2*D3) =INDEX(MINVERSE(G2:H3), 2, 1) =INDEX(MINVERSE(G2:H3), 2, 2) =INDEX(MINVERSE(I2:J3), 2, 1) =INDEX(MINVERSE(I2:J3), 2, 2) =INDEX(MMULT(M2:N3, G2:H3), 2, 1) + INDEX(MMULT(K2:L3, I2:J3), 2, 1) =INDEX(MMULT(M2:N3, G2:H3), 2, 2) + INDEX(MMULT(K2:L3, I2:J3), 2, 2) =INDEX(MINVERSE(O2:P3), 2, 1) =INDEX(MINVERSE(O2:P3), 2, 2) =INDEX(MMULT(M2:N3, E2:E3), 2, 1) + INDEX(MMULT(K2:L3, F2:F3), 2, 1) =INDEX(MMULT(M2:N3, F2:F3), 2, 1) - INDEX(MMULT(K2:L3, E2:E3), 2, 1) =INDEX(MMULT(Q2:R3, S2:S3), 2, 1) =INDEX(MMULT(Q2:R3, T2:T3), 2, 1) 3
4 vector X vector Y R THETA vector Fr vector Fi matrix Jr matrix Ji matrix 1/Jr matrix 1/Ji matrix Marie = 1/Ji*Jr + 1/Jr*Ji matrix 1/Marie vector Vince = 1/Ji*Fr + 1/Jr*Fi vector Vito = 1/Ji*Fi - 1/Jr*Fr vector DX = 1/Marie * Vince vector DY = 1/Marie * Vito 4
5 = A2 - U2 = B2 - V2 =SQRT(A5^2 + B5^2) =IF(B5<0, -ACOS(A5/C5), ACOS(A5/C5)) = C5^3*C6^5*COS(3*D5 + 5*D6) - 6*C5*C6*COS(D5 + D6) - 17 = C5^3*C6^5*SIN(3*D5 + 5*D6) - 6*C5*C6*SIN(D5 + D6) = 3*C5^2*C6^5*COS(2*D5 + 5*D6) - 6*C6*COS(D6) = 5*C5^3*C6^4*COS(3*D5 + 4*D6) - 6*C5*COS(D5) = 3*C5^2*C6^5*SIN(2*D5 + 5*D6) - 6*C6*SIN(D6) = 5*C5^3*C6^4*SIN(3*D5 + 4*D6) - 6*C5*SIN(D5) =INDEX(MINVERSE(G5:H6), 1, 1) =INDEX(MINVERSE(G5:H6), 1, 2) =INDEX(MINVERSE(I5:J6), 1, 1) =INDEX(MINVERSE(I5:J6), 1, 2) =INDEX(MMULT(M5:N6, G5:H6), 1, 1) + INDEX(MMULT(K5:L6, I5:J6), 1, 1) =INDEX(MMULT(M5:N6, G5:H6), 1, 2) + INDEX(MMULT(K5:L6, I5:J6), 1, 2) =INDEX(MINVERSE(O5:P6), 1, 1) =INDEX(MINVERSE(O5:P6), 1, 2) =INDEX(MMULT(M5:N6, E5:E6), 1, 1) + INDEX(MMULT(K5:L6, F5:F6), 1, 1) =INDEX(MMULT(M5:N6, F5:F6), 1, 1) - INDEX(MMULT(K5:L6, E5:E6), 1, 1) =INDEX(MMULT(Q5:R6, S5:S6), 1, 1) =INDEX(MMULT(Q5:R6, T5:T6), 1, 1) 5
6 = A3 - U3 = B3 - V3 =SQRT(A6^2 + B6^2) =IF(B6<0, -ACOS(A6/C6), ACOS(A6/C6)) = 7*C5^2*C6*COS(2*D5 + D6) - 3*C5*C6^3*COS(D5 + 3*D6) + 5 = 7*C5^2*C6*SIN(2*D5 + D6) - 3*C5*C6^3*SIN(D5 + 3*D6) = 14*C5*C6*COS(D5 + D6) - 3*C6^3*COS(3*D6) = 7*C5^2*COS(2*D5) - 9*C5*C6^2*COS(D5 + 2*D6) = 14*C5*C6*SIN(D5 + D6) - 3*C6^3*SIN(3*D6) = 7*C5^2*SIN(2*D5) - 9*C5*C6^2*SIN(D5 + 2*D6) =INDEX(MINVERSE(G5:H6), 2, 1) =INDEX(MINVERSE(G5:H6), 2, 2) =INDEX(MINVERSE(I5:J6), 2, 1) =INDEX(MINVERSE(I5:J6), 2, 2) =INDEX(MMULT(M5:N6, G5:H6), 2, 1) + INDEX(MMULT(K5:L6, I5:J6), 2, 1) =INDEX(MMULT(M5:N6, G5:H6), 2, 2) + INDEX(MMULT(K5:L6, I5:J6), 2, 2) =INDEX(MINVERSE(O5:P6), 2, 1) =INDEX(MINVERSE(O5:P6), 2, 2) =INDEX(MMULT(M5:N6, E5:E6), 2, 1) + INDEX(MMULT(K5:L6, F5:F6), 2, 1) =INDEX(MMULT(M5:N6, F5:F6), 2, 1) - INDEX(MMULT(K5:L6, E5:E6), 2, 1) =INDEX(MMULT(Q5:R6, S5:S6), 2, 1) =INDEX(MMULT(Q5:R6, T5:T6), 2, 1) 6
7 vector X vector Y R THETA vector Fr vector Fi matrix Jr matrix Ji matrix 1/Jr matrix 1/Ji matrix Marie = 1/Ji*Jr + 1/Jr*Ji matrix 1/Marie vector Vince = 1/Ji*Fr + 1/Jr*Fi vector Vito = 1/Ji*Fi - 1/Jr*Fr vector DX = 1/Marie * Vince vector DY = 1/Marie * Vito 7
8 = A5 - U5 = B5 - V5 =SQRT(A8^2 + B8^2) =IF(B8<0, -ACOS(A8/C8), ACOS(A8/C8)) = C8^3*C9^5*COS(3*D8 + 5*D9) - 6*C8*C9*COS(D8 + D9) - 17 = C8^3*C9^5*SIN(3*D8 + 5*D9) - 6*C8*C9*SIN(D8 + D9) = 3*C8^2*C9^5*COS(2*D8 + 5*D9) - 6*C9*COS(D9) = 5*C8^3*C9^4*COS(3*D8 + 4*D9) - 6*C8*COS(D8) = 3*C8^2*C9^5*SIN(2*D8 + 5*D9) - 6*C9*SIN(D9) = 5*C8^3*C9^4*SIN(3*D8 + 4*D9) - 6*C8*SIN(D8) =INDEX(MINVERSE(G8:H9), 1, 1) =INDEX(MINVERSE(G8:H9), 1, 2) =INDEX(MINVERSE(I8:J9), 1, 1) =INDEX(MINVERSE(I8:J9), 1, 2) =INDEX(MMULT(M8:N9, G8:H9), 1, 1) + INDEX(MMULT(K8:L9, I8:J9), 1, 1) =INDEX(MMULT(M8:N9, G8:H9), 1, 2) + INDEX(MMULT(K8:L9, I8:J9), 1, 2) =INDEX(MINVERSE(O8:P9), 1, 1) =INDEX(MINVERSE(O8:P9), 1, 2) =INDEX(MMULT(M8:N9, E8:E9), 1, 1) + INDEX(MMULT(K8:L9, F8:F9), 1, 1) =INDEX(MMULT(M8:N9, F8:F9), 1, 1) - INDEX(MMULT(K8:L9, E8:E9), 1, 1) =INDEX(MMULT(Q8:R9, S8:S9), 1, 1) =INDEX(MMULT(Q8:R9, T8:T9), 1, 1) 8
9 = A6 - U6 = B6 - V6 =SQRT(A9^2 + B9^2) =IF(B9<0, -ACOS(A9/C9), ACOS(A9/C9)) = 7*C8^2*C9*COS(2*D8 + D9) - 3*C8*C9^3*COS(D8 + 3*D9) + 5 = 7*C8^2*C9*SIN(2*D8 + D9) - 3*C8*C9^3*SIN(D8 + 3*D9) = 14*C8*C9*COS(D8 + D9) - 3*C9^3*COS(3*D9) = 7*C8^2*COS(2*D8) - 9*C8*C9^2*COS(D8 + 2*D9) = 14*C8*C9*SIN(D8 + D9) - 3*C9^3*SIN(3*D9) = 7*C8^2*SIN(2*D8) - 9*C8*C9^2*SIN(D8 + 2*D9) =INDEX(MINVERSE(G8:H9), 2, 1) =INDEX(MINVERSE(G8:H9), 2, 2) =INDEX(MINVERSE(I8:J9), 2, 1) =INDEX(MINVERSE(I8:J9), 2, 2) =INDEX(MMULT(M8:N9, G8:H9), 2, 1) + INDEX(MMULT(K8:L9, I8:J9), 2, 1) =INDEX(MMULT(M8:N9, G8:H9), 2, 2) + INDEX(MMULT(K8:L9, I8:J9), 2, 2) =INDEX(MINVERSE(O8:P9), 2, 1) =INDEX(MINVERSE(O8:P9), 2, 2) =INDEX(MMULT(M8:N9, E8:E9), 2, 1) + INDEX(MMULT(K8:L9, F8:F9), 2, 1) =INDEX(MMULT(M8:N9, F8:F9), 2, 1) - INDEX(MMULT(K8:L9, E8:E9), 2, 1) =INDEX(MMULT(Q8:R9, S8:S9), 2, 1) =INDEX(MMULT(Q8:R9, T8:T9), 2, 1) 9

SHEET 4B. Values returned by the above functions.
  A B C D E F G H I J K L M N O P Q R S T U V  
1 vector X vector Y R THETA vector Fr vector Fi matrix Jr matrix Ji matrix 1/Jr matrix 1/Ji matrix Marie = 1/Ji*Jr + 1/Jr*Ji matrix 1/Marie vector Vince = 1/Ji*Fr + 1/Jr*Fi vector Vito = 1/Ji*Fi - 1/Jr*Fr vector DX = 1/Marie * Vince vector DY = 1/Marie * Vito 1
2 1 -2 2.2360680 -1.1071487 25 -34 78 214 18 -28 0.0052 -0.0743 0.0084 -0.0236 3.6119 2.0446 0.3711 0.1075 1.6116 -2.0405 0.5110 -0.5571 2
3 1 -1 1.4142136 -0.7853982 -26 -13 -8 15 -36 -10 0.0028 0.0271 -0.0303 -0.0152 -3.1668 -7.0605 -0.1664 -0.1898 -0.8099 1.8616 -0.1145 -0.0138 3
4 vector X vector Y R THETA vector Fr vector Fi matrix Jr matrix Ji matrix 1/Jr matrix 1/Ji matrix Marie = 1/Ji*Jr + 1/Jr*Ji matrix 1/Marie vector Vince = 1/Ji*Fr + 1/Jr*Fi vector Vito = 1/Ji*Fi - 1/Jr*Fr vector DX = 1/Marie * Vince vector DY = 1/Marie * Vito 4
5 0.489014027 -1.442856945 1.5234733 -1.2440256 0.6379 -10.1120 43.3812 78.6293 14.6679 -20.8000 0.0125 -0.0682 -0.0084 -0.0532 1.6139 -1.9179 0.3654 -0.1464 0.2766 -0.1108 0.1025 -0.1585 5
6 1.114485968 -0.986232727 1.4881982 -0.7244218 -4.6227 -2.3883 -6.6887 14.4616 -21.1175 3.2955 0.0058 0.0376 -0.0540 -0.0375 -2.8018 -4.7870 -0.2139 -0.1232 -0.0095 0.8061 -0.0580 -0.0756 6
7 vector X vector Y R THETA vector Fr vector Fi matrix Jr matrix Ji matrix 1/Jr matrix 1/Ji matrix Marie = 1/Ji*Jr + 1/Jr*Ji matrix 1/Marie vector Vince = 1/Ji*Fr + 1/Jr*Fi vector Vito = 1/Ji*Fi - 1/Jr*Fr vector DX = 1/Marie * Vince vector DY = 1/Marie * Vito 7
8 0.386532948 -1.284334111 1.3412389 -1.2784592 -1.2703 -1.9698 28.4282 55.1863 21.4933 -3.5676 0.0179 -0.0804 0.0797 0.0419 3.7608 4.3037 0.3737 -0.1119 -0.1493 -0.1855 -8.82E-03 -2.54E-02 8
9 1.172485927 -0.910628350 1.4845765 -0.6603492 -0.5669 -0.1354 -6.1139 12.2847 -17.0086 6.7837 0.0089 0.0414 0.1998 0.2525 3.6237 14.3777 -0.0942 0.0977 -0.4201 -0.3930 -2.70E-02 -2.09E-02 9
10 vector X vector Y R THETA vector Fr vector Fi matrix Jr matrix Ji matrix 1/Jr matrix 1/Ji matrix Marie = 1/Ji*Jr + 1/Jr*Ji matrix 1/Marie vector Vince = 1/Ji*Fr + 1/Jr*Fi vector Vito = 1/Ji*Fi - 1/Jr*Fr vector DX = 1/Marie * Vince vector DY = 1/Marie * Vito 10
11 0.395349375 -1.258978595 1.3195940 -1.2665237 -0.1496 0.0794 25.6986 54.6416 25.9262 3.6068 0.0193 -0.0888 0.0299 -0.0135 2.8242 0.8344 0.3672 -0.0620 -4.35E-03 3.21E-03 -9.89E-04 6.18E-04 11
12 1.199483241 -0.889685430 1.4934190 -0.6381834 -2.02E-02 1.90E-02 -5.6747 11.8796 -16.6583 7.9595 0.0092 0.0418 0.0625 0.0973 0.5979 4.9379 -0.0445 0.2100 -9.80E-03 9.04E-03 -7.86E-03 1.76E-03 12
13 vector X vector Y R THETA vector Fr vector Fi matrix Jr matrix Ji matrix 1/Jr matrix 1/Ji matrix Marie = 1/Ji*Jr + 1/Jr*Ji matrix 1/Marie vector Vince = 1/Ji*Fr + 1/Jr*Fi vector Vito = 1/Ji*Fi - 1/Jr*Fr vector DX = 1/Marie * Vince vector DY = 1/Marie * Vito 13
14 0.396338818 -1.259596682 1.3204803 -1.2659491 7.09E-04 -4.08E-04 26.0099 55.1444 26.1455 3.6287 0.0192 -0.0885 0.0297 -0.0135 2.8247 0.8368 0.3673 -0.0623 1.55E-05 -1.51E-05 2.62E-06 -3.00E-06 14
15 1.201347402 -0.891440836 1.4959620 -0.6383835 1.14E-04 -4.34E-05 -5.6635 11.9609 -16.6775 8.0034 0.0091 0.0417 0.0618 0.0969 0.6008 4.9352 -0.0447 0.2102 4.93E-05 -4.06E-05 9.68E-06 -7.87E-06 15
16 vector X vector Y R THETA vector Fr vector Fi matrix Jr matrix Ji matrix 1/Jr matrix 1/Ji matrix Marie = 1/Ji*Jr + 1/Jr*Ji matrix 1/Marie vector Vince = 1/Ji*Fr + 1/Jr*Fi vector Vito = 1/Ji*Fi - 1/Jr*Fr vector DX = 1/Marie * Vince vector DY = 1/Marie * Vito 16
17 0.396336198 -1.259593684 1.3204767 -1.2659503 1.50E-08 -1.04E-08 26.0084 55.1420 26.1445 3.6287 0.0192 -0.0885 0.0297 -0.0135 2.8247 0.8368 0.3673 -0.0623 2.50E-10 -3.49E-10 2.47E-11 -7.01E-11 17
18 1.201337727 -0.891432968 1.4959496 -0.6383831 2.71E-09 -4.55E-10 -5.6635 11.9606 -16.6775 8.0031 0.0091 0.0417 0.0618 0.0969 0.6008 4.9352 -0.0447 0.2102 1.08E-09 -9.34E-10 2.15E-10 -1.81E-10 18
19 vector X vector Y R THETA vector Fr vector Fi matrix Jr matrix Ji matrix 1/Jr matrix 1/Ji matrix Marie = 1/Ji*Jr + 1/Jr*Ji matrix 1/Marie vector Vince = 1/Ji*Fr + 1/Jr*Fi vector Vito = 1/Ji*Fi - 1/Jr*Fr vector DX = 1/Marie * Vince vector DY = 1/Marie * Vito 19
20 0.396336198 -1.259593684 1.3204767 -1.2659503 0 2.13E-14 26.0084 55.1420 26.1445 3.6287 0.0192 -0.0885 0.0297 -0.0135 2.8247 0.8368 0.3673 -0.0623 7.04E-16 6.77E-16 2.55E-16 1.87E-16 20
21 1.201337727 -0.891432968 1.4959496 -0.6383831 0 -3.33E-15 -5.6635 11.9606 -16.6775 8.0031 0.0091 0.0417 0.0618 0.0969 0.6008 4.9352 -0.0447 0.2102 5.47E-17 9.95E-16 -2.00E-17 1.79E-16 21
  A B C D E F G H I J K L M N O P Q R S T U V  

Index

URL http://members.aceweb.com/patrussell/approximations/Sheet_4.htm
Unpublished Work. © Copyright 2001 Pat Russell. Updated April 17, 2009.