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**Multi-Dimensional Arrays**

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**Matrices Arrays seen so far are single dimensional**

A linear list of items One subscript for accessing individual elements Multidimensional arrays are useful organizations in many applications values of temperatures at different locations in different times number of runs in different sessions on different days in a test match trajectory of a planet in space - time Matrices simplest and useful multidimensional arrays Two dimensions

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**Two-Dimensional Data Structures**

Fortran allows variables that can assume two dimensional array values Typical values are: true. .true. false. .false. true. .true. An integer 3 by 3 matrix: 3 rows and 3 columns of integer values 4 by 3 real matrix 3 by 2 logical matrix

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**Matrix Declaration Matrix entries can be of different types**

No. of rows and columns can be arbitrary Fortran Declaration defines these two attributes for a matrix variable Real, Dimension(3,6):: Sum Logical, Dimension(23:100,-2:5):: flag integer, Dimension(15:19,5):: runs Accessing an element Sum(2,5), flag(56,0), runs(18,3) Two subscripts are needed

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**Matrix Representation**

Memory is Single-Dimensional Linear representation of matrix elements Row major order: One row after another Column major order: one column after another (assumed representation) Example: 15 20 3 56 54 45 67 77 86 76 11 22

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**Initialization/Assignment**

Using array constructor (column major order) A = (\ 2,3,4,3,4,5,4,5,6 \) Explicit nested do loops do I = 1, 5 do J = 1,8 A(I,J) = I + J end do Whole row/column assignment do I =1,5 A(I,:) = I Initialization can be done at the time of declaration

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**Assignment Whole, part or individual array elements can be assigned**

A(:,i) = ... A(i,j) = … The rhs should be appropriate type Lifting of scalar to suitable types may be required with READ statements read *, ((A(i,j),j = 1, 3), I = 1, 4)

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**Using matrix variables**

Matrix variables can be accessed, individually, in sections or as a whole Used in expressions on the right hand side of any assignment Intrinsic functions on base types lifted to matrix types Eg. Abs, Sin, Cos, Tan, Log, Sqrt can be applied to a matrix variable Sin(A) is a similar matrix in which each entry is Sin of the corresponding entries in A

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**Other intrinsic functions**

LBOUND(A,d) – lower bound of index in the dimension d LBOUND(A) - list of lower bounds UBOUND - upper bound, similar SIZE(A,d) - number of entries in dimension d size(A) - total no. of elements SHAPE(A) - returns the `shape' of A Extent in a dimension is the no. of entries in that dimension Shape is an one dimensional array of extents(one entry per dimension)

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**Transformational Intrinsic Functions**

Dot_Product(A,B) - dot product of one dimensional arrays of A and B MATMUL(A,B) - Matrix multiplication on conformable matrices A and B RESHAPE(A,B) - B is a one-dimensional shape array Entries in A are converted into an array of shape B Column major order used Example: Suppose A = and B = [6,2] Reshape(A,B) = Usually used for initializing a matrix from a linear list of elements

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Masked Assignment A more generalized selection is possible using WHERE construct Suppose, you want to multiply all entries > 0 by 2, and all other entries set to 0 then where (A > 0) A = 2 * A elsewhere A = 0 end where Note: Reference to an array has many connotations – context dependent

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**Accessing Matrix variables**

When a matrix is accessed, the index values should be valid illegal values lead to run-time error - Array index exceeding bounds Compiler optionally inserts code for the check

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**Generalized Arrays Multi-dimensional arrays are possible**

All array operations, like declaration, initialization and use are similar Rank of an array is the number of dimensions One-dimensional array is of rank 1, matrix of rank 2 etc. Arrays of up to rank 7 are allowed in Fortran

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**Memory Allocation Arrays are allocated memory by the compiler**

The allocation is static - done at compile time More and larger arrays, larger the memory requirement Array use and their size requirements should be done with care indiscriminate use would result in wastage of memory Use arrays only when all the data in the arrays are required at the same time in memory

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Allocatable Arrays Allocatable arrays is a solution to memory wastage problem When the size of an input data set is unknown, better to use the following: real, dimension(), allocatable :: A integer, dimension(:), allocatable :: B This is a declaration of an Array A and B of rank 1 and 2 respectively Only ranks specified, the extent decided dynamically When A ( and B) initialized, the extents are decided and allocated This enables avoiding static allocation of conservative estimate of required space

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Allocate Construct Allocatable arrays are allocated memory using explicit instruction: allocate(A(n), i) This indicates the extent of A is the value of n This is an external routine that may succeed or fail Succeeds when the memory allocation is successful The external procedure sends a value 0 via the parameter i Typically the size is input by the user which is used in the allocation Allocated memory can be deallocated using deallocate(A,i) This succeeds provided the value returned via i is 0

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**Illustration Linear Systems of Equations**

Most commonly occurring problem, the original motivation for matrices solve a system of n equations in n variables a11x1 + a12x a1nxn = b1 a12x2 + a22x a2nxn = b2 ... an1x1 + an2x2 + … + annxn = bn aij,, bj are all real numbers

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Gaussian Elimination choose any equation and any variable with non zero coefficient ( called pivot) in the equation eliminate this variable from all other equations by adding a multiple of chosen equation repeat with remaining equations solve one equation in one variable back substitute value of variable to find others two steps, elimination and back substitution

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Program real, dimension(:,:), allocatable :: a integer :: n, i, j real, dimension(:), allocatable :: x read *, n ! Number of variables allocate(x(n), stat=i) if ( i /= 0) then print *, "allocation of x failed" stop endif allocate(a(n,n+1), stat=i) print *, "allocation of array a failed" do i = 1,n print *, "enter coefficients of equation", i read *, a(i, 1:n+1) end do

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do i = 1,n a(i, i:n+1) = a(i,i:n+1)/a(i,i) ! make coefficient of x_i 1.0 do j = i+1,n a(j,i:n+1) = a(j,i:n+1) - a(i,i:n+1)*a(j,i) end do ! eliminate x_i end do do i = n, 1, -1 x(i) = a(i,n+1) - sum(a(i,i+1:n)*x(i+1:n)) end do ! sum of a null array is 0.0 print *, "solution is" print *, x(1:n) deallocate(a, x, stat=i) if ( i /= 0) then print *, "deallocation failed" stop endif

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Elimination a(1:n+1,1:n+1) contains the coefficients, x(1:n) the computed values of variables assumes a(i,i) /= 0.0 at every step true for many special types of matrices problem if abs(a(i,i)) is too small elimination needs to be done only once for different right hand sides only back substitution needs to be done reduces operations from n3 to n2

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**Pivoting swap rows and columns to make a(i,i) large**

partial pivoting - swap row i with row containing maxval(abs(a(i:n,i))) what if it is 0.0? complete pivoting- swap row and column to make a(i,i) = maxval(abs(a(i:n,i:n))) what if it is still 0.0? this changes order of the variables gives more accurate results but more time

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