1 | -- Hier comes further description due to M2: |
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4 | jacobian(Matrix) -- the matrix of partial derivatives of polynomials in a matrix |
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5 | Synopsis |
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6 | Usage: jacobian f |
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7 | |
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8 | Function: jacobian |
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9 | Inputs: |
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10 | f, a matrix, with one row -- veraltet???? |
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11 | Outputs: |
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12 | a matrix, the Jacobian matrix of partial derivatives of the polynomial entries of f |
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13 | Description |
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14 | If f is a 1 by m matrix over a polynomial ring R with n indeterminates, then the resulting matrix of partial derivatives has dimensions n by m, and the (i,j) entry is the partial derivative of the j-th entry of f by the i-th indeterminate of the ring. |
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16 | If the ring of f is a quotient polynomial ring S/J, then only the derivatives of the given entries of f are computed and NOT the derivatives of elements of J.i1 : R = QQ[x,y,z]; |
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21 | i2 : f = matrix{{y^2-x*(x-1)*(x-13)}} |
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22 | |
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23 | o2 = | -x3+14x2+y2-13x | |
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25 | 1 1 |
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26 | o2 : Matrix R <--- R |
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27 | i3 : jacobian f |
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28 | |
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29 | o3 = {1} | -3x2+28x-13 | |
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30 | {1} | 2y | |
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31 | {1} | 0 | |
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32 | |
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33 | 3 1 |
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34 | o3 : Matrix R <--- R |
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39 | If the ring of f is a polynomial ring over a polynomial ring, then indeterminates in the coefficient ring are treated as constants.i4 : R = ZZ[a,b,c][x,y,z] |
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43 | |
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44 | o4 = R |
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45 | |
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46 | o4 : PolynomialRing |
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47 | i5 : jacobian matrix{{a*x+b*y^2+c*z^3, a*x*y+b*x*z}} |
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48 | |
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49 | o5 = {1} | a ya+zb | |
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50 | {1} | 2yb xa | |
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51 | {1} | 3z2c xb | |
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52 | |
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53 | 3 2 |
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54 | o5 : Matrix R <--- R |
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55 | |
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