data

1ecece86-742b-4e70-aa0b-3da72c018712 14 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

x y f1 4 x 2^* 7 y 5^* - 12 x cos * - 0 + :

t0 1 time :

x y f2 4 x 2^* 7 y 5^* - 12 x cos * - 5 + :

bound the hi/lo limits lo 5 - : hi 5 : N 100 : 13 mat

level 0102031 mat :

populate solutions U lo hi lo hi lo - N / + (range :

Iterative "roots" solve iso-contour k C Typical Roots range ± 5 j U length 1 range rho j el sol U j el y f1 level k el - y roots at : for U rho augment 31 line :

1 C 2 C 3 C 1.1 0 + 125 black 1 5 mat 4 1 sys $data 1 C eval :$ $1 time t0 - min 3.3$ $Finite differences derivative Y 0 : X data 1 col : Y data 2 col : 12 mat n X rows : deriv 1 el 0 : deriv n el 0 : i 2 n 1 - (range deriv i el 0.5 Y i 1 - el Y i el - X i 1 - el X i el - / Y i el Y i 1 - el - X i el X i 1 - el - / + (* : for deriv 1 el deriv 2 el : deriv n el deriv deriv rows 1 - el : D X deriv augment : Arc integrand .................... xx D 1 col : δx D 2 col : 12 mat i 1 δx rows range yy i el 1 δx i el (2^+ sqrt eval : 11 line for Arc xx yy augment : Arc length ....................... vx Arc 1 col : vy Arc 2 col : 12 mat S 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : S i el S i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for Σ vx S augment : 191 line$

Derivative of the function

Arc integrand

Arc

Arc length ...

Σ Σ rows row 5 13.3094 12 mat

Finite differences Area vx data 1 col : vy data 2 col : 12 mat S 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : S i el S i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for Σ vx S augment : 51 line

Cumulative Area ...

vx data 1 col : vy data 2 col : 12 mat x f vx vy x cinterp : 21 mat

data 1 row data 51 row Σ Σ rows row 31 mat 5 - 1.6903 12 mat 0 1.1138 - 12 mat 5 8.589 12 mat 31 mat

acc 100 x f x 5 - 5 int 8.589

data

data 5 - 1.6538 4.9 - 1.643 4.8 - 1.6323 4.7 - 1.6217 4.6 - 1.6111 4.5 - 1.6006 4.4 - 1.59 4.3 - 1.5793 4.2 - 1.5684 4.1 - 1.5574 4 - 1.5461 3.9 - 1.5345 3.8 - 1.5224 3.7 - 1.5099 3.6 - 1.4969 3.5 - 1.4832 3.4 - 1.4687 3.3 - 1.4535 3.2 - 1.4373 3.1 - 1.4201 3 - 1.4017 2.9 - 1.382 2.8 - 1.3608 2.7 - 1.3379 2.6 - 1.3131 2.5 - 1.2859 2.4 - 1.2561 2.3 - 1.223 2.2 - 1.1859 2.1 - 1.1437 2 - 1.0945 1.9 - 1.0351 1.8 - 0.9593 1.7 - 0.85 1.6 - 0.6098 1.5557 - 0 1.5 - 0.7662 - 1.4 - 0.9029 - 1.3 - 0.9838 - 1.2 - 1.0417 - 1.1 - 1.0866 - 1 - 1.1227 - 0.9 - 1.1523 - 0.8 - 1.1768 - 0.7 - 1.1972 - 0.6 - 1.2141 - 0.5 - 1.2278 - 0.4 - 1.2387 - 0.3 - 1.247 - 0.2 - 1.2528 - 0.1 - 1.2562 - 11015 -^* - 1.2574 - 0.1 1.2562 - 0.2 1.2528 - 0.3 1.247 - 0.4 1.2387 - 0.5 1.2278 - 0.6 1.2141 - 0.7 1.1972 - 0.8 1.1768 - 0.9 1.1523 - 1 1.1227 - 1.1 1.0866 - 1.2 1.0417 - 1.3 0.9838 - 1.4 0.9029 - 1.5 0.7662 - 1.5557 0 1.6 0.6098 1.7 0.85 1.8 0.9593 1.9 1.0351 2 1.0945 2.1 1.1437 2.2 1.1859 2.3 1.223 2.4 1.2561 2.5 1.2859 2.6 1.3131 2.7 1.3379 2.8 1.3608 2.9 1.382 3 1.4017 3.1 1.4201 3.2 1.4373 3.3 1.4535 3.4 1.4687 3.5 1.4832 3.6 1.4969 3.7 1.5099 3.8 1.5224 3.9 1.5345 4 1.5461 4.1 1.5574 4.2 1.5684 4.3 1.5793 4.4 1.59 4.5 1.6006 4.6 1.6111 4.7 1.6217 4.8 1.6323 4.9 1.643 5 1.6538 1032 mat :

vx 1.6 - 1.5 - 21 mat :

vy 0.6098 0.7662 - 21 mat :

x data 1 col : y data 2 col (: 12 mat

Area Polygon wrt vertices coordinates x y A 12 / x i el y i 1 + el * x i 1 + el y i el * - (i 1 x rows 1 - sum * abs :

Perimeter  Polygon wrt vertices coordinates x y P x i 1 + el x i el - (2^y i 1 + el y i el - (2^+ sqrt i 1 x rows 1 - sum :

Center of gravity x y Xo 16 x y A * / x i el x i 1 + el + (x i el y i 1 + el * x i 1 + el y i el * - (* i 1 x rows 1 - sum * : x y Yo 16 x y A * / y i el y i 1 + el + (x i el y i 1 + el * x i 1 + el y i el * - (* i 1 x rows 1 - sum * : 21 mat

Center of gravity x y Xo abs x y Yo abs 21 mat 2.8677 1016 -^* 1.8435 21 mat

Area
Perimeter x y A x y P 21 mat 2.2597 13.4763 21 mat

13.3094

Xo 0 : Yo x y Yo abs x f x 5 - 5 int / : 21 sys 0 0.2146 21 sys

Notes ... 1. Area & Perimeter look OK 2. Xo, Yo ... uncertified

data Xo Yo . 14 red 1 5 mat 2 1 sys

Procedure to add the two missing Y(0)

Data ... data

Data 5 - 1.6538 4.9 - 1.643 4.8 - 1.6323 4.7 - 1.6217 4.6 - 1.6111 4.5 - 1.6006 4.4 - 1.59 4.3 - 1.5793 4.2 - 1.5684 4.1 - 1.5574 4 - 1.5461 3.9 - 1.5345 3.8 - 1.5224 3.7 - 1.5099 3.6 - 1.4969 3.5 - 1.4832 3.4 - 1.4687 3.3 - 1.4535 3.2 - 1.4373 3.1 - 1.4201 3 - 1.4017 2.9 - 1.382 2.8 - 1.3608 2.7 - 1.3379 2.6 - 1.3131 2.5 - 1.2859 2.4 - 1.2561 2.3 - 1.223 2.2 - 1.1859 2.1 - 1.1437 2 - 1.0945 1.9 - 1.0351 1.8 - 0.9593 1.7 - 0.85 1.6 - 0.6098 1.5 - 0.7662 - 1.4 - 0.9029 - 1.3 - 0.9838 - 1.2 - 1.0417 - 1.1 - 1.0866 - 1 - 1.1227 - 0.9 - 1.1523 - 0.8 - 1.1768 - 0.7 - 1.1972 - 0.6 - 1.2141 - 0.5 - 1.2278 - 0.4 - 1.2387 - 0.3 - 1.247 - 0.2 - 1.2528 - 0.1 - 1.2562 - 11015 -^* - 1.2574 - 0.1 1.2562 - 0.2 1.2528 - 0.3 1.247 - 0.4 1.2387 - 0.5 1.2278 - 0.6 1.2141 - 0.7 1.1972 - 0.8 1.1768 - 0.9 1.1523 - 1 1.1227 - 1.1 1.0866 - 1.2 1.0417 - 1.3 0.9838 - 1.4 0.9029 - 1.5 0.7662 - 1.6 0.6098 1.7 0.85 1.8 0.9593 1.9 1.0351 2 1.0945 2.1 1.1437 2.2 1.1859 2.3 1.223 2.4 1.2561 2.5 1.2859 2.6 1.3131 2.7 1.3379 2.8 1.3608 2.9 1.382 3 1.4017 3.1 1.4201 3.2 1.4373 3.3 1.4535 3.4 1.4687 3.5 1.4832 3.6 1.4969 3.7 1.5099 3.8 1.5224 3.9 1.5345 4 1.5461 4.1 1.5574 4.2 1.5684 4.3 1.5793 4.4 1.59 4.5 1.6006 4.6 1.6111 4.7 1.6217 4.8 1.6323 4.9 1.643 5 1.6538 1012 mat :

data 5 - 1.6538 4.9 - 1.643 4.8 - 1.6323 4.7 - 1.6217 4.6 - 1.6111 4.5 - 1.6006 4.4 - 1.59 4.3 - 1.5793 4.2 - 1.5684 4.1 - 1.5574 4 - 1.5461 3.9 - 1.5345 3.8 - 1.5224 3.7 - 1.5099 3.6 - 1.4969 3.5 - 1.4832 3.4 - 1.4687 3.3 - 1.4535 3.2 - 1.4373 3.1 - 1.4201 3 - 1.4017 2.9 - 1.382 2.8 - 1.3608 2.7 - 1.3379 2.6 - 1.3131 2.5 - 1.2859 2.4 - 1.2561 2.3 - 1.223 2.2 - 1.1859 2.1 - 1.1437 2 - 1.0945 1.9 - 1.0351 1.8 - 0.9593 1.7 - 0.85 1.6 - 0.6098 1.5557 - 0 1.5 - 0.7662 - 1.4 - 0.9029 - 1.3 - 0.9838 - 1.2 - 1.0417 - 1.1 - 1.0866 - 1 - 1.1227 - 0.9 - 1.1523 - 0.8 - 1.1768 - 0.7 - 1.1972 - 0.6 - 1.2141 - 0.5 - 1.2278 - 0.4 - 1.2387 - 0.3 - 1.247 - 0.2 - 1.2528 - 0.1 - 1.2562 - 11015 -^* - 1.2574 - 0.1 1.2562 - 0.2 1.2528 - 0.3 1.247 - 0.4 1.2387 - 0.5 1.2278 - 0.6 1.2141 - 0.7 1.1972 - 0.8 1.1768 - 0.9 1.1523 - 1 1.1227 - 1.1 1.0866 - 1.2 1.0417 - 1.3 0.9838 - 1.4 0.9029 - 1.5 0.7662 - 1.5557 0 1.6 0.6098 1.7 0.85 1.8 0.9593 1.9 1.0351 2 1.0945 2.1 1.1437 2.2 1.1859 2.3 1.223 2.4 1.2561 2.5 1.2859 2.6 1.3131 2.7 1.3379 2.8 1.3608 2.9 1.382 3 1.4017 3.1 1.4201 3.2 1.4373 3.3 1.4535 3.4 1.4687 3.5 1.4832 3.6 1.4969 3.7 1.5099 3.8 1.5224 3.9 1.5345 4 1.5461 4.1 1.5574 4.2 1.5684 4.3 1.5793 4.4 1.59 4.5 1.6006 4.6 1.6111 4.7 1.6217 4.8 1.6323 4.9 1.643 5 1.6538 1032 mat :

data Procedure to add the missing y0 points

vx 1.6 - 1.5 - 21 mat :

vy 0.6098 0.7662 - 21 mat :

x f vx vy x linterp :

x f 0 ≡ x 1.6 - 1.5 - solve 1.5557 -

u 1.5557 - 012 mat :

v 1.5557 012 mat :

M B r InsertRow a M 1 r 1 M cols submatrix : c M r 1 + M rows 1 M cols submatrix : a B c stack 31 line :

idx 1 Data rows range : N idx Data augment : 21 sys

New Data u 35 InsertRow : M New v 67 InsertRow : 21 sys

N 11 mat 15 - 1.6538 2 4.9 - 1.643 3 4.8 - 1.6323 4 4.7 - 1.6217 5 4.6 - 1.6111 6 4.5 - 1.6006 7 4.4 - 1.59 8 4.3 - 1.5793 9 4.2 - 1.5684 10 4.1 - 1.5574 114 - 1.5461 12 3.9 - 1.5345 13 3.8 - 1.5224 14 3.7 - 1.5099 15 3.6 - 1.4969 16 3.5 - 1.4832 17 3.4 - 1.4687 18 3.3 - 1.4535 19 3.2 - 1.4373 20 3.1 - 1.4201 213 - 1.4017 22 2.9 - 1.382 23 2.8 - 1.3608 24 2.7 - 1.3379 25 2.6 - 1.3131 26 2.5 - 1.2859 27 2.4 - 1.2561 28 2.3 - 1.223 29 2.2 - 1.1859 30 2.1 - 1.1437 312 - 1.0945 32 1.9 - 1.0351 33 1.8 - 0.9593 34 1.7 - 0.85 35 1.6 - 0.6098 36 1.5 - 0.7662 - 37 1.4 - 0.9029 - 38 1.3 - 0.9838 - 39 1.2 - 1.0417 - 40 1.1 - 1.0866 - 411 - 1.1227 - 42 0.9 - 1.1523 - 43 0.8 - 1.1768 - 44 0.7 - 1.1972 - 45 0.6 - 1.2141 - 46 0.5 - 1.2278 - 47 0.4 - 1.2387 - 48 0.3 - 1.247 - 49 0.2 - 1.2528 - 50 0.1 - 1.2562 - 5111015 -^* - 1.2574 - 52 0.1 1.2562 - 53 0.2 1.2528 - 54 0.3 1.247 - 55 0.4 1.2387 - 56 0.5 1.2278 - 57 0.6 1.2141 - 58 0.7 1.1972 - 59 0.8 1.1768 - 60 0.9 1.1523 - 611 1.1227 - 62 1.1 1.0866 - 63 1.2 1.0417 - 64 1.3 0.9838 - 65 1.4 0.9029 - 66 1.5 0.7662 - 67 1.6 0.6098 68 1.7 0.85 69 1.8 0.9593 70 1.9 1.0351 712 1.0945 72 2.1 1.1437 73 2.2 1.1859 74 2.3 1.223 75 2.4 1.2561 76 2.5 1.2859 77 2.6 1.3131 78 2.7 1.3379 79 2.8 1.3608 80 2.9 1.382 813 1.4017 82 3.1 1.4201 83 3.2 1.4373 84 3.3 1.4535 85 3.4 1.4687 86 3.5 1.4832 87 3.6 1.4969 88 3.7 1.5099 89 3.8 1.5224 90 3.9 1.5345 914 1.5461 92 4.1 1.5574 93 4.2 1.5684 94 4.3 1.5793 95 4.4 1.59 96 4.5 1.6006 97 4.6 1.6111 98 4.7 1.6217 99 4.8 1.6323 100 4.9 1.643 1015 1.6538 1013 mat 11 mat

New 11 mat 5 - 1.6538 4.9 - 1.643 4.8 - 1.6323 4.7 - 1.6217 4.6 - 1.6111 4.5 - 1.6006 4.4 - 1.59 4.3 - 1.5793 4.2 - 1.5684 4.1 - 1.5574 4 - 1.5461 3.9 - 1.5345 3.8 - 1.5224 3.7 - 1.5099 3.6 - 1.4969 3.5 - 1.4832 3.4 - 1.4687 3.3 - 1.4535 3.2 - 1.4373 3.1 - 1.4201 3 - 1.4017 2.9 - 1.382 2.8 - 1.3608 2.7 - 1.3379 2.6 - 1.3131 2.5 - 1.2859 2.4 - 1.2561 2.3 - 1.223 2.2 - 1.1859 2.1 - 1.1437 2 - 1.0945 1.9 - 1.0351 1.8 - 0.9593 1.7 - 0.85 1.6 - 0.6098 1.5557 - 0 1.5 - 0.7662 - 1.4 - 0.9029 - 1.3 - 0.9838 - 1.2 - 1.0417 - 1.1 - 1.0866 - 1 - 1.1227 - 0.9 - 1.1523 - 0.8 - 1.1768 - 0.7 - 1.1972 - 0.6 - 1.2141 - 0.5 - 1.2278 - 0.4 - 1.2387 - 0.3 - 1.247 - 0.2 - 1.2528 - 0.1 - 1.2562 - 11015 -^* - 1.2574 - 0.1 1.2562 - 0.2 1.2528 - 0.3 1.247 - 0.4 1.2387 - 0.5 1.2278 - 0.6 1.2141 - 0.7 1.1972 - 0.8 1.1768 - 0.9 1.1523 - 1 1.1227 - 1.1 1.0866 - 1.2 1.0417 - 1.3 0.9838 - 1.4 0.9029 - 1.5 0.7662 - 1.6 0.6098 1.7 0.85 1.8 0.9593 1.9 1.0351 2 1.0945 2.1 1.1437 2.2 1.1859 2.3 1.223 2.4 1.2561 2.5 1.2859 2.6 1.3131 2.7 1.3379 2.8 1.3608 2.9 1.382 3 1.4017 3.1 1.4201 3.2 1.4373 3.3 1.4535 3.4 1.4687 3.5 1.4832 3.6 1.4969 3.7 1.5099 3.8 1.5224 3.9 1.5345 4 1.5461 4.1 1.5574 4.2 1.5684 4.3 1.5793 4.4 1.59 4.5 1.6006 4.6 1.6111 4.7 1.6217 4.8 1.6323 4.9 1.643 5 1.6538 1022 mat 11 mat

M 11 mat 5 - 1.6538 4.9 - 1.643 4.8 - 1.6323 4.7 - 1.6217 4.6 - 1.6111 4.5 - 1.6006 4.4 - 1.59 4.3 - 1.5793 4.2 - 1.5684 4.1 - 1.5574 4 - 1.5461 3.9 - 1.5345 3.8 - 1.5224 3.7 - 1.5099 3.6 - 1.4969 3.5 - 1.4832 3.4 - 1.4687 3.3 - 1.4535 3.2 - 1.4373 3.1 - 1.4201 3 - 1.4017 2.9 - 1.382 2.8 - 1.3608 2.7 - 1.3379 2.6 - 1.3131 2.5 - 1.2859 2.4 - 1.2561 2.3 - 1.223 2.2 - 1.1859 2.1 - 1.1437 2 - 1.0945 1.9 - 1.0351 1.8 - 0.9593 1.7 - 0.85 1.6 - 0.6098 1.5557 - 0 1.5 - 0.7662 - 1.4 - 0.9029 - 1.3 - 0.9838 - 1.2 - 1.0417 - 1.1 - 1.0866 - 1 - 1.1227 - 0.9 - 1.1523 - 0.8 - 1.1768 - 0.7 - 1.1972 - 0.6 - 1.2141 - 0.5 - 1.2278 - 0.4 - 1.2387 - 0.3 - 1.247 - 0.2 - 1.2528 - 0.1 - 1.2562 - 11015 -^* - 1.2574 - 0.1 1.2562 - 0.2 1.2528 - 0.3 1.247 - 0.4 1.2387 - 0.5 1.2278 - 0.6 1.2141 - 0.7 1.1972 - 0.8 1.1768 - 0.9 1.1523 - 1 1.1227 - 1.1 1.0866 - 1.2 1.0417 - 1.3 0.9838 - 1.4 0.9029 - 1.5 0.7662 - 1.5557 0 1.6 0.6098 1.7 0.85 1.8 0.9593 1.9 1.0351 2 1.0945 2.1 1.1437 2.2 1.1859 2.3 1.223 2.4 1.2561 2.5 1.2859 2.6 1.3131 2.7 1.3379 2.8 1.3608 2.9 1.382 3 1.4017 3.1 1.4201 3.2 1.4373 3.3 1.4535 3.4 1.4687 3.5 1.4832 3.6 1.4969 3.7 1.5099 3.8 1.5224 3.9 1.5345 4 1.5461 4.1 1.5574 4.2 1.5684 4.3 1.5793 4.4 1.59 4.5 1.6006 4.6 1.6111 4.7 1.6217 4.8 1.6323 4.9 1.643 5 1.6538 1032 mat 11 mat