Arithmetic Array Operators

Arithmetic operators on arrays and certain combinations of arrays and scalars are available through the standard arithmetic operators, addition (+), subtraction (-), division (/), multiplication as well as through the exponentiation (^) operator.

1 Addition and Subtraction

Strangely enough, elementwise operations for the combination of an array and a scalar operand are defined for multiplication and division but not for addition and subtraction.

class AddSub Real Add1 = {1, 2, 3} + 1; // Not allowed! Real Add2 = {1, 2, 3} + {1, 2}; // Not allowed, different array sizes! Real Add3[2, 2] = {{1, 1}, {2, 2}} + {{1, 2}, {3, 4}}; // Result: {{2, 3}, {5, 6}} Real Sub1[3] = {1, 2, 3} - {1, 2, 0}; // Result: {0, 0, 3} end AddSub;
class AddSub2 Real Add3[2, 2] = {{1, 1}, {2, 2}} + {{1, 2}, {3, 4}}; // Result: {{2, 3}, {5, 6}} Real Sub1[3] = {1, 2, 3} - {1, 2, 0}; // Result: {0, 0, 3} end AddSub2;

Try interactive:

{{1, 1}, {2, 2}} // Works
[1, 1; 2, 2] //Works
2.0*{{1, 1}, {2, 2}} // Works
{{1, 1}, {2, 2}} + {{1, 2}, {3, 4}} // Works

1.1 Simulation of AddSub and AddSub2

When we try to simulate AddSub, it does not work since different array sizes are not allowed in Addition operand

simulate( AddSub )
simulate( AddSub2 )

Look at results:

val(Sub1[1],1.)
plot({Sub1[1], Sub1[3],Add3[1,1]})

2 Division

class Div Real[3] Div1, Div2, Div3; equation Div1 = {2, 4, 6} / 2; // Result: {1, 2, 3} Div2 = 6 / {1, 2, 3}; // Not allowed! Div3 = {1, 2, 3} / {1, 2, 2}; // Not allowed! end Div;
class Div2 Real Div1[3]; equation Div1 = {2, 4, 6} / 2; // Result: {1, 2, 3} end Div2;

2.1 Simulation of Div and Div2

simulate( Div ) //Error in model
simulate( Div2 )
val(Div1[2],0.)
plot({Div1[1],Div1[2],Div1[3]})

3 Multiplication

The array multiplication operator is only defined for vectors and matrices and has several different definitions depending on the type of its arguments. If both operands are vectors then array multiplication means scalar product, otherwise, if the operands are vectors or matrices then it means matrix multiplication. If one of the operands is a scalar, then the multiplication operation is defined as elementwise multiplication, which however is not available when both operands are arrays despite the fact that analogous elementwise operations are defined for addition and subtraction. The reason is that we have already used the multiplication notation (*) for the more common operation of matrix multiplication.

class Multiplication Real m1[3] = {1, 2, 3} * 2; // Elementwise mult: {2, 4, 6}; Real m2[3] = 3 * {1, 2, 3}; // Elementwise mult: {3, 6, 9}; Real m3 = {1, 2, 3} * {1, 2, 2}; // Scalar product: 11; Real m4[2] = {{1, 2}, {3, 4}} * {1, 2}; // Matrix mult: {5, 11}; Real m5[1] = {1, 2, 3} * {{1}, {2}, {10}}; // Matrix mult: {35}; Real m6[1] = {1, 2, 3} * [1; 2; 10]; // Matrix mult: {35}; Real m7[2, 2] = {{1, 2}, {3, 4}} * {{1, 2}, {2, 1}}; // Matrix mult: {{5, 4}, {11, 10}}; Real m8[2, 2] = [1, 2; 3, 4] * [1, 2; 2, 1]; // Matrix mult: {{5, 4}, {11, 10}}; end Multiplication;
{{1, 2}, {3, 4}} * {1, 2}
{1, 2, 3} * [1; 2; 10]

3.1 Simulation of Multiplication

simulate( Multiplication )
val(m1[2],0.)
val(m3,0.)
plot({m1[2],m3})

4 Exponentiation

Array exponentiation is the same as repeated matrix multiplication and is therefore only defined when the first operand is a square matrix and the second operand is a non-negative integer.

class Exp Real e1[2, 2]; Real e2[2, 2]; equation e1 = {{1, 2}, {1, 2}} ^ 0; // Result: {{1, 0}, {0, 1}} e2 = [1, 2; 1, 2] ^ 2; // Result: {{3, 6}, {3, 6}} end Exp;

4.1 Simulation of Exponentiation

simulate( Exp )
val(e1[1,1],0)
plot({e1[1,1],e2[1,2]})