APMonitor Summation with Vectors

Variable or object arrays are defined by square brackets with a range of integers and separated by a colon as variable[index 1:index 2]. Arrays may be used to define multiple equations or connections on one line. Any line with an array is processed sequentially from the lowest to the highest index. The model parser creates and processes the arrays as if they were written sequential in non-array form as shown in the example.

Higher dimensional arrays

Arrays with more than one dimension are allowed. The array indices are separated by brackets as var[i][j][k]. For operations on matrices, the precedence of operations is determined by the number of colons separating the vector indices. Matrix elements with fewer colon separators are executed first. For example, a set of 24 intermediate variables posed as:

x[1:2][1::3][1:::4] = 1

results in the following set of equations:

x[1][1][1] = 1
x[2][1][1] = 1
x[1][2][1] = 1
x[2][2][1] = 1
x[1][3][1] = 1
x[2][3][1] = 1
x[1][1][2] = 1
etc...

Array Index Consistency

When processing the arrays, the parser checks for array size consistency. An error with an appropriate message is returned if the vector indeces are of different dimension.

Examples

 ! Method #1: Summation with arrays
 Model array
   Constants
      n = 5
   End Constants

   Parameters
     p[1:n] = 1
   End Parameters

   Variables
     sum
   End Variables

   Intermediates
     z[1] = p[1]
     z[2:n] = z[1:n-1] + p[2:n]
   End Intermediates

   Equations
     sum = z[n]
   End Equations
 End Model

 ! Method #2: Summation with the sum object
 Objects
   z = sum(5)
 End Objects

 Connections
   p[1:n] = z.x[1:n]
   y = z.y
 End Connections

 Model
   Constants
      n = 5
   End Constants

   Parameters
     p[1:n] = 1
   End Parameters

   Variables
     y
     sum
   End Variables

   Equations
     sum = y
   End Equations
 End Model

 ! Method #3: Summation without arrays
 Model array
   Parameters
     p[1] = 1
     p[2] = 1
     p[3] = 1
     p[4] = 1
     p[5] = 1
   End Parameters

   Variables
     sum
   End Variables

   Intermediates
     z[1] = p[1]
     z[2] = z[1] + p[2]
     z[3] = z[2] + p[3]
     z[4] = z[3] + p[4]
     z[5] = z[4] + p[5]
   End Intermediates

   Equations
     sum = z[5]
   End Equations
 End Model

An additional example is a matrix summation where there are two indices of the parameter matrix p.

 ! Matrix Summation
 Model
   Parameters
     p[1:10][1::5] = 1
   End Parameters

   Variables
     x
   End Variables

   Intermediates
     ! sum the rows
     n[0][1:5] = 0
     n[1:10][1::5] = n[0:9][1::5] + p[1:10][1::5]
     ! sum the columns that are summation of rows
     m[0] = 0
     m[1:5] = m[0:4] + n[10][1:5]
   End Intermediates

   Equations
     ! solution = 50
     x = m[5]
   End Equations
 End Model

Arrays in Python GEKKO

Multi-dimensional arrays are defined in Python GEKKO with the m.Array() function or with list comprehensions.

from gekko import GEKKO

m = GEKKO()

ni = 3 # number of rows
nj = 2 # number of columns

# best method: use m.Array function
x = m.Array(m.Var,(ni,nj))
m.Equations([x[i][j]==i*j+1 for i in range(ni) for j in range(nj)])

# another way: list comprehensions
y = [[m.Var() for j in range(nj)] for i in range(ni)]
for i in range(ni):
for j in range(nj):
m.Equation(x[i][j]**2==y[i][j])

# summation
z = m.Var()
m.Equation(z==sum([sum([x[i][j] for i in range(ni)]) for j in range(nj)]))

m.solve()

print('x:')
print(x)
print('y=x**2:')
print(y)
print('z')
print(z.value)

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APMonitor