MATRIX A = 0 MATRIX A = 1

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resets all A[i, k] = 0 produces a unit matrix A (1s along diagonal)

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Programs with Vector/Matrix Operations and Submodels

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MATRIX B = $In(A) MATRIX B = A% MATRIX D = a * A * B * C *

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makes B the matrix inverse of A (if it exists) makes B the matrix transpose of A produces a matrix product D (a is an optional scalar)

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These assignments return error messages if matrices are not square or unconformable, or if an inverse does not exist As noted, for properly dimensioned rectangular matrices A, B

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MATRIX B = A% makes B the transpose of A (b[i, k] = a[k, i] for all i, k)

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Nonconformable matrices A, B are again rejected with an error message 3-10 Matrix Assignments and Difference Equations in DYNAMIC Program Segments DYNAMIC program segments can manipulate matrices declared in the experiment protocol with matrix assignments

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MATRIX W = matrix expression

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Matrix expressions are functions of scalars a, b, , vectors u, v, , and/or matrices A, B, , which can be constants or variables Some examples are

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MATRIX W = a * A + b MATRIX W = a * A + b * B MATRIX W = recip(A) MATRIX W = sin(A) MATRIX W = u * v MATRIX W = u & v ( W[i, k] = a * A[i,k] + b ) ( W[i, k] = a * A[i,k] + b * B[i, k] ) ( W[i, k] = 1/A[i, k] ) ( W[i, k] = sin(A[i, k]) ) ( W[i, k] = u[i] v[k] ) ( W[i, k] = min{u[i], v [k])})

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The syntax of more general matrix expressions is defined in the DESIRE reference manual Matrices can, moreover, be manipulated as equivalent vectors (Section 3-11) Matrix difference equations are used mainly to modify matrix-vector products W * x in optimization studies (control systems, statistical regression, model matching, and neural networks) In particular, the matrix difference equation

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DELTA W = matrix expression

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is equivalent to

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MATRIX W = W + matrix expression

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The resulting matrix elements W[i, k] are difference-equation state variables (Section 2-2) Their initial values default to zero unless otherwise specified by the experiment protocol They are not reset by reset or drunr statements The precautions of Section 2-2 apply 3-11 Vector and Matrix Operations using Equivalent Vectors DESIRE experiment protocol scripts can use two very useful equivalence declarations similar to those in Fortran In particular, the modified ARRAY declaration

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ARRAY x1[n1] + x2[n2] + = x,

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declares concatenated subvectors x1, x2, together with a vector x of dimension n1 + n2 + whose elements overlay the subvectors x1, x2, , starting with x1 One can then access, say, x2[3] also as x[n1+3] Subvectors are particularly useful in neural-network simulations (Section 6-2) The second type of equivalence declaration

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ARRAY V[n, m] = v

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allows one to access a two-dimensional array and its elements both as an n m matrix V and as a vector v with dimension nm Then equivalent vector expressions with the convenient Vector and Vectr d/dt operations to relate and modify matrices can be used This technique can often (not always) replace matrix assignments Applications include image processing, fuzzy-logic models (Section 7-7), and landscape modeling (Section 7-15) Note that both concatenated subvectors and equivalent array vectors allow identification of maximum and minimum elements of large arrays by the method of Section 3-8

VECTORS IN PHYSICS AND CONTROL-SYSTEM PROBLEMS 3-12 Vectors in Physics Problems Vectors such as forces or velocities are not just useful shorthand notations for multiple equations; they are intuitively meaningful abstractions Many relations used in physics problems are most easily understood when we represent them in vector form, for example,

Vectr d/dt position = velocity | Vectr d/dt velocity = force/mass

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However, to obtain numerical results such as trajectory plots, it is usually necessary to specify vector components and initial values as scalar subscripted variables 3-13 Simulation of a Nuclear Reactor The program in Figure 3-2 shows a compact vector model of the chain reaction in a nuclear reactor D Hetrick s classical textbook problem [1 3] lumps the entire reactor into a single core region and neglects chain-reaction poisoning by reaction products such as xenon The state variables are the normalized chain-reaction power output enp (proportional to neutron density), the reactor temperature temprtr, and six normalized precursor-product densities d[1], d[2], , d[6] Our compact vector model collects the state variables d[i] into a six-dimensional state vector d When the control-rod input b * t increases the reactivity r, the chain reaction increases enp dramatically In the educational TRIGA reactor, the resulting increase in the reactor temperature in turn reduces the reactivity r, so that a short and safe power pulse results (Fig 3-2b) 3-14 Linear Transformations and Rotation Matrices Simple vector assignments such as Vector y = A * x conveniently implement linear operations on vectors, such as rotations Note that y = A * x can represent the result of rotating the vector x into a new position, or y may be a representation of x in a rotated coordinate system The rotation of a plane vector x (x[1], x[2]) into the vector y (y[1], y[2]) can be programmed with two scalar defined-variable assignments

y[1] = x[1] * cos(theta) - x[2] * sin(theta) y[2] = x[1] * sin(theta) + x[2] * cos(theta)

Instead, a two-dimensional rotation matrix A with ARRAY A[2, 2] can be declared in the experiment protocol, and then possibly time-variable elements A[i, k] of A are specified in a DYNAMIC program segment:

A[1,1] = cos(theta) | A[1,2] = sin (theta) A[2,1] = A[1,2] | A[2,2] = A[1,1]