~v200 200 ~w175 5 391 744 0 737 0 0 ~f? 14 12 10 ? 2 1 1 0 ? ? ? "Arial" ? ? ? 1 ? 0 1 "Times" 12 ? ? 5 0 c n 106 1 0 0 k 468 i"?n page ?p?a" -2 1 26177 26178 26115 26178 1 1 1 1 0 0 8405120 0 -1 0 1 -1 -1 -1 -1 -1 0 1 ? ? ~Q ]|Expr|[#b @`bb#_b#_b#_}))# b'4" *|: ;bP:&c0!)Least Squares Polynomial| | Curve Fitting}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 0 ~V?m0 (a)~p0 1 ~V?m0 (A)~p0 1 ~V?m0 (X)~p0 1 ~V?m0 (Y)~p0 1 ~V?f0 (g)~p0 1 ~V?f144 (RowsOf)~p0 1 ~V?v0 (k)~p0 1 ~V?v0 (y)~p0 1 ~V?v0 (x)~p0 1 ~V?c1 ('p)~p0 1 ~Q ]|Expr|[#b @`bb#_b#_b#_}`fb#L})!# b'4`f }["! `fb#L}) # b'4| |}& b!( b"0 b#8 b$@ b%H b&P!WW})!# b'4`fb#@}" *|: ;bP8&c0!*Data| |}& b!( b"0 b#8 b$@ b%H b&P!WW}}}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0 0 ~A(X=(0;1;1.5;2.2999999999999998;2.5;4;5.0999999999999996;6;6.5;~ 7;8.0999999999999996;9;9.3000000000000007;11;11.300000000000001;~ 12.1;13.1;14;15.5;16;17.5;17.800000000000001;19;20))~p0 1 ~d~A(Y=(0.20000000000000001;0.80000000000000004;2.5;2.5;3.5;4.2999999999999998;~ 3;5;3.5;2.3999999999999999;1.3;2;-0.29999999999999999;-1.3;-3;~ -4;-4.9000000000000004;-4;-5.2000000000000002;-3;-3.5;-1.6000000000000001;~ -1.3999999999999999;-0.10000000000000001))~p0 255 ~d~Q ]|Expr|[#b @`bb#_b#_b#_})b"5# b#\" *|: ;bP8&c0!*The data appear| | to have a mininmum near x ,] 15 and a maximum near x ,] 6,N | | The lowest order polynomial that can reproduce such behavior| | is a cubic,N Thus the basis functions are those monomials ,H| | 1,L x,L $^x_^2,L $^x_^3 ,I that span the cubics,N The algorithm| | may be applied to other bases ,H $^g^1_,L $^g^2_,L ,N,N,N ,L| | $^g^k_ ,I with attendant modification of matrix A,N| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0 255 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b$@["! ) # b$@| |}& b!( b"0 b#8 b$@ b%H b&P!WW})## b$@" *|: ;bP8&c0!*Normal equation| |}& b!( b"0 b#8 b$@ b%H b&P!WW}}}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0 0 ~A(A^**A*a=A^**Y)~p0 1 ~A(a=(A^**A)^(-1)*A^**Y)~p0 255 ~sb/_!! } )+! c#T" c"H#"c'8 _c/__c/__! !} ^ _~Q ]|Expr|[#b @`bb#_b#_b#_})## b$@" *|: ;bP8&c0!*matrix A| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 1 ~A(A=((g(X_1))_1,(g(X_1))_2,(g(X_1))_3,(g(X_1))_4;(g(X_2))_1,~ (g(X_2))_2,(g(X_2))_3,(g(X_2))_4;(g(X_3))_1,(g(X_3))_2,(g(X_3))_~ 3,(g(X_3))_4;(g(X_4))_1,(g(X_4))_2,(g(X_4))_3,(g(X_4))_4;(g(X_~ 5))_1,(g(X_5))_2,(g(X_5))_3,(g(X_5))_4;(g(X_6))_1,(g(X_6))_2,~ (g(X_6))_3,(g(X_6))_4;(g(X_7))_1,(g(X_7))_2,(g(X_7))_3,(g(X_7))_~ 4;(g(X_8))_1,(g(X_8))_2,(g(X_8))_3,(g(X_8))_4;(g(X_9))_1,(g(X_~ 9))_2,(g(X_9))_3,(g(X_9))_4;(g(X_10))_1,(g(X_10))_2,(g(X_10))_~ 3,(g(X_10))_4;(g(X_11))_1,(g(X_11))_2,(g(X_11))_3,(g(X_11))_4;~ (g(X_12))_1,(g(X_12))_2,(g(X_12))_3,(g(X_12))_4;(g(X_13))_1,(~ g(X_13))_2,(g(X_13))_3,(g(X_13))_4;(g(X_14))_1,(g(X_14))_2,(g(~ X_14))_3,(g(X_14))_4;(g(X_15))_1,(g(X_15))_2,(g(X_15))_3,(g(X_~ 15))_4;(g(X_16))_1,(g(X_16))_2,(g(X_16))_3,(g(X_16))_4;(g(X_17))_~ 1,(g(X_17))_2,(g(X_17))_3,(g(X_17))_4;(g(X_18))_1,(g(X_18))_2,~ (g(X_18))_3,(g(X_18))_4;(g(X_19))_1,(g(X_19))_2,(g(X_19))_3,(~ g(X_19))_4;(g(X_20))_1,(g(X_20))_2,(g(X_20))_3,(g(X_20))_4;(g(~ X_21))_1,(g(X_21))_2,(g(X_21))_3,(g(X_21))_4;(g(X_22))_1,(g(X_~ 22))_2,(g(X_22))_3,(g(X_22))_4;(g(X_23))_1,(g(X_23))_2,(g(X_23))_~ 3,(g(X_23))_4;(g(X_24))_1,(g(X_24))_2,(g(X_24))_3,(g(X_24))_4))~p0 2 ~d~Q ]|Expr|[#b @`bb#_b#_b#_})## b!L" *|: ;bP8&c0!*basis functions| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0 1 ~A(g(?x)=(1;?x;?x^2;?x^3))~p0 2 ~d~A(a=(-0.35934718246510433;2.3051112162430734;-0.35319013922840842;~ 0.01206019829786123))~p0 255 ~d~sb/_!! } $&! c#T"!c"H#_c/__c/__} ^ _~Q ]|Expr|[#b @`bb#_b#_b#_}`f#})!# b!L`f }["! `f#}) # b!L| |}& b!( b"0 b#8 b$@ b%H b&P!WW})%# b!L" *|: ;bP8&c0!*Least squares| | fit}& b!( b"0 b#8 b$@ b%H b&P!WW}}}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0 0 ~A(y=Summation(k):(1):(RowsOf(a))*(a_k*(g(x))_k))~p0 255 ~d~G1 1 268 496 0 2 4 4 10 (-1...21):(-6...6):(?=0...2*'p):('p/~ 5):(10)~Q ]|Expr|[#b @`bb#_b#_b#_})!# b#@" *|: ;bP8&c0!*Declarations| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 0 ~R8405120 ? (x,y):(y=bottom...top):(x=left...right):(0)~p0 1 ~gc1 2 ? 0 -3724 -4096 0 65535 -3724 4096 0 ~gc1 2 ? 0 -1862 -4096 0 65535 -1862 4096 0 ~gc1 2 ? 0 0 -4096 0 65535 0 4096 0 ~gc1 2 ? 0 1862 -4096 0 65535 1862 4096 0 ~gc1 2 ? 0 3724 -4096 0 65535 3724 4096 0 ~ ~R8405120 ? (x,y):(x=left...right):(y=bottom...top):(0)~p0 1 ~gc1 2 ? 0 -4096 -2731 0 65535 4096 -2731 0 ~gc1 2 ? 0 -4096 -1365 0 65535 4096 -1365 0 ~gc1 2 ? 0 -4096 0 0 65535 4096 0 0 ~gc1 2 ? 0 -4096 1365 0 65535 4096 1365 0 ~gc1 2 ? 0 -4096 2731 0 65535 4096 2731 0 ~ ~X2 8405120 (left,y):(y=bottom...top):(y)~p0 1 ~gc1 2 ? 0 -4096 -4096 0 65535 -4096 4096 0 ~X1 8405120 (x,bottom):(~ x=left...right):(x)~p0 1 ~gc1 2 ? 0 -4096 -4096 0 65535 4096 -4096 0 ~V?c64 (left)~p0 1 ~V?c65 (right)~p0 1 ~V?c66 (bottom)~p0 1 ~V?c67 (top)~p0 1 ~L1 255 ? (x,y):(x=left...right)~p0 0 ~gc0 56 ? 0 -4096 -2068 0 1170 -3950 -1292 0 3120 -3706 -171 0 4030 -3592 280 0 5201 -3446 798 0 6241 -3316 1200 0 7411 -3170 1589 0 8321 -3056 1849 0 9362 -2926 2101 0 10272 -2812 2283 0 11442 -2666 2468 0 12482 -2536 2588 0 13653 -2389 2676 0 14221 -2318 2702 0 14434 -2292 2709 0 14563 -2276 2712 0 15603 -2146 2720 0 16513 -2032 2699 0 17684 -1885 2638 0 18724 -1755 2552 0 19894 -1609 2423 0 20804 -1495 2302 0 21845 -1365 2141 0 22755 -1252 1984 0 26135 -829 1285 0 27046 -715 1072 0 31207 -195 21 0 32377 -49 -285 0 33287 65 -523 0 37448 585 -1564 0 38618 731 -1831 0 40569 975 -2238 0 41479 1089 -2408 0 42649 1235 -2605 0 43690 1365 -2757 0 44860 1512 -2900 0 45770 1625 -2989 0 46810 1755 -3065 0 47151 1798 -3083 0 47720 1869 -3106 0 48451 1960 -3122 0 49281 2064 -3120 0 49931 2145 -3104 0 51101 2292 -3037 0 52011 2405 -2952 0 53052 2536 -2817 0 53962 2649 -2664 0 55132 2796 -2418 0 56172 2926 -2149 0 57343 3072 -1788 0 58253 3186 -1463 0 59293 3316 -1041 0 60203 3429 -628 0 61374 3576 -31 0 64364 3950 1840 0 65535 4096 2717 0 ~ ~S17 ? 16711680 ? ? (X_k,Y_k):(k=1...RowsOf(X)):(6)~p0 0 ~gc-1 24 ? 0 -3724 137 0 3205 -3351 546 0 5698 -3165 1707 0 8548 -2867 1707 0 11041 -2793 2389 0 14246 -2234 2935 0 17096 -1825 2048 0 20301 -1489 3413 0 22794 -1303 2389 0 25644 -1117 1638 0 28137 -707 887 0 31342 -372 1365 0 34192 -261 -205 0 37397 372 -887 0 39890 484 -2048 0 42740 782 -2731 0 45233 1154 -3345 0 48438 1489 -2731 0 51288 2048 -3550 0 54493 2234 -2048 0 56986 2793 -2389 0 59836 2904 -1092 0 62329 3351 -956 0 65535 3724 -68 0 ~t~p0 0 ~c1 17 -1 16 -1 ~c9 22 -1 17 -1 13 -1 4 -1 12 -1 3 -1 21 -1 5 -1 19 -1 2 -1 ~e