Multidimensional Arrays

1
Multidimensional Arrays

• Multidimensional array is the array with two or
  more dimensions.
• For example
• char box 3 3 defines a two-dimensional array
  
• and
• box21 is an element in row 2 , column 1
• and
• char box3 can be used in the function
  prototype
• note that only the first dimension can be omitted
  

2
Multidimensional Arrays

• For example
• double table NROWSNCOLS
• Can be used as parameter in the function
  prototype as
• void
• process_martrix( int in 4,
• int out 4,
• int nrows)

3
Two Dimensional Array

• Char box 3 3

Column
0
1
2
Row
0
box 1 2
1
2
4

• /
• Checks whether a box is completely filled
• /
• int
• filled(char box33) / input - box to
  check /
• int r,c, / row and column subscripts /
• ans / whether or not box is
  filled. /
• / Assumes box is filled until blank is
  found /
• ans 1
• / Resets ans to zero if a blank is
  found /
• for (r 0 r lt 3 r)
• for (c 0 c lt 3 c)
• if (boxrc ' ')
• ans 0
• return (ans)

5
Arrays with Several Dimensions

• int soil_type4 7 MAXDEPTH

6
Case Study Cellular Telephone System

• Problem Finding the best way to build a cellular
  network. There is some marketing data that
  predicts the demand will be at tree time of
  interest. There are only 10 transmitters and
  there is a need for a program to help analyzing
  call demand data.

7
Case Study Cellular Telephone System

• Analysis There should be three matrices shows
  traffic density for each time of the day
• Input
• int commutersGRID_SIZEGRID_SIZE
• int salesforceGRID_SIZEGRID_SIZE
• int weekendGRID_SIZEGRID_SIZE
• Output
• int summed_dataGRID_SIZEGRID_SIZE
• int location_i, location_j

8
Case Study Cellular Telephone System

• Design initial algorithm
• Get traffic data for three time period
• Get the weights from user
• Multiply weight by each matrix entry and store
  the sum in the summed data
• Find highest valued cells in the summed data and
  display them as the pair of location_i and
  location_j
• Implementation

9
Filling the multidimensional array

• / Fills 3 GRID_SIZE x GRID_SIZE arrays with
  traffic data from TRAFFIC_FILE/
• void
• get_traffic_data(int commutersGRID_SIZEGRID_SIZ
  E, / output /
• int salesforceGRID_SIZEGRID_SI
  ZE, / output /
• int weekendGRID_SIZEGRID_SIZE
  ) / output /
• int i, j / loop counters /
• FILE fp / file pointer /
• fp fopen(TRAFFIC_FILE, "r")
• for (i 0 i lt GRID_SIZE i)
• for (j 0 j lt GRID_SIZE j)
• fscanf(fp, "d", commutersij)
• for (i 0 i lt GRID_SIZE i)
• for (j 0 j lt GRID_SIZE j)
• fscanf(fp, "d", salesforceij)
• for (i 0 i lt GRID_SIZE i)
• for (j 0 j lt GRID_SIZE j)
• fscanf(fp, "d", weekendij)
• fclose(fp)

10
Modifying the multidimensional array

• / Computes and displays the weighted,
  summed_data /
• for (i 0 i lt GRID_SIZE i)
• for (j 0 j lt GRID_SIZE j)
• summed_dataij
  commuter_weight
• 
  commutersij
• 
  salesforce_weight
• 
  salesforceij
• 
  weekend_weight
• 
  weekendij

11
Searching in the multidimensional array

• / Finds the NUM_TRANSMITTERS highest values in
  the summed_data matrix.Temporarily stores the
  coordinates in location_i and location_j, and
  then displays the resulting locations /
• printf("\n\nLocations of the d
  transmitters\n\n", NUM_TRANSMITTERS)
• for (tr 1 tr lt NUM_TRANSMITTERS
  tr)
• current_max SELECTED / Starts off
  our search with a
• value
  that is known to be too low. /
• for (i 0 i lt GRID_SIZE i)
• for (j 0 j lt GRID_SIZE j)
• if (current_max lt
  summed_dataij)
• current_max
  summed_dataij
• location_i i
• location_j j

12
Printing the contents of multidimensional array

• /
• Displays contents of a GRID_SIZE x GRID_SIZE
  matrix of integers
• /
• void
• print_matrix(int matrixGRID_SIZEGRID_SIZE)
• int i, j / loop counters /
• for (i 0 i lt GRID_SIZE i)
• for (j 0 j lt GRID_SIZE j)
• printf("3d ", matrixij)
• printf("\n")

13
Vectors

• Vector a mathematical object consisting of a
  sequence of numbers.
• / a vector lt4, 12, 19gt /
• int vect3 4, 12, 19
• Differences between vector and array
• 1- an n_dimensional vector is represented in C
  as a one dimensional array of size n.
• 2- vect3 is vect2 in C

14
Vectors

• Calculating scalar product
• lt1,2,4gt. lt2,3,1gt 12 23 4112
• In C
• sum_prod 0
• for (k0 kltn k)
• sum_prod xk wk

15
Matrices

• Matrix a mathematical object consisting of a
  rectangular arrangement of numbers called the
  element of matrix..
• / a matrix 3 6
• 4 5
• int x22 3, 6, 4, 5

3
6
x
4
5
16
Matrices

• Multiplying a matrix by a vector
• A X V
• 1 1 1 5
• 2 3 1 1 10
  multiplication
• 1 -1 -1 2 -3
  on
• 0 1 2 2 6
  the right
• In C for each member of V
• vi 0
• for (k0 kltn k)
• vk aik xk

17

• / Computes the product of M-by-N matrix a and
  the N-dimensional vector x. The result is stored
  in the output parameter v, an M-dimensional
  vector./
• void mat_vec_prod(double v, / M-dimensional
  vector /
• double aMN, /
  M-by-N matrix /
• double x) /
  N-dimensional vector /
• int i, k
• for (i 0 i lt M i)
• vi 0
• for (k 0 k lt N k)
• vi aik xk

18
Matrix Multiplication

• 1 1 1 2 0 1 6 0 0
• 2 3 1 1 -1 0 10 -2 1
  
• 1 -1 -1 3 1 -1 -2 0 2
• for ( i0 ilt m , i)
• for (j0 jltp j)
• .. compute cij.

19

• / Multiplies matrices A and B yielding product
  matrix C /
• void mat_prod(double cMP, / output - M by
  P matrix /
• double aMN, / input - M by
  N matrix /
• double bNP) / input - N by
  P matrix /
• int i, j, k
• for (i 0 i lt M i)
• for (j 0 j lt P j)
• cij 0
• for (k 0 k lt N k)
• cij aik
  bkj

20
Solving System of Linear Equations

• To solve many problems such as force equation in
  three-dimensional system we need to solve a three
  linear equations
• A X
  Y
• 1 1 1 x1
  4
• 2 3 1 x2
  9
• 1 -1 -1 x3
  -2
• It is multiplication of a matrix by a vector on
  the right

21
Gaussian Elimination

• Gaussian elimination can be used to solve a
  linear equation.
• The algorithm for Gussian elimination is
• Transform the original system into scaled
  triangular form.
• Solve for xi by back substitution

22
Gaussian Elimination

• triangular form
• 1 1 1 x1
  4
• 0 1 -1 x2
  1
• 0 0 1 x3
  1
• back substitution
• x1 x2 x3 4
• x2 - x3 1
• x3 1

23
Gaussian Elimination

• For doing that we need to triangularizing the
  augmented matrix by following operations
• Multiply any row of aug by nonzero number
• Add to any row of aug a multiple of other rows
• Swap any two rows
• If system has a unique solution, we can get the
  system into desired form by this three
  operations.

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• /
• Performs pivoting with respect to the pth row
  and the pth column
• If no nonzero pivot can be found, FALSE is
  sent back through piv_foundp
• /
• void pivot(double augNN1, / input/output -
  augmented matrix /
• int p, / input - current
  row /
• int piv_foundp) / output - whether or
  not nonzero pivot found /
• double xmax, xtemp
• int j, k, max_row
• / Finds maximum pivot /
• xmax fabs(augpp)
• max_row p
• for (j p1 j lt N j)
• if (fabs(augjp) gt xmax)
• xmax fabs(augjp)
• max_row j

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• / Swaps rows if nonzero pivot was found /
• if (xmax 0)
• piv_foundp FALSE
• else
• piv_foundp TRUE
• if (max_row ! p) / swap rows /
• for (k p k lt N1 k)
• xtemp augpk
• augpk
  augmax_rowk
• augmax_rowk xtemp

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• /
• Performs back substitution to compute a
  solution vector to a system of
• linear equations represented by the augmented
  matrix aug. Assumes that
• the coefficient portion of the augmented
  matrix has been triangularized,
• and its diagonal values are all 1.
• /
• void
• back_sub(double augNN1, / input - scaled,
  triangularized
• 
  augmented matrix /
• double xN) / output -
  solution vector /
• double sum
• int i, j
• xN - 1 augN - 1N
• for (i N - 2 i gt 0 --i)
• sum 0
• for (j i 1 j lt N j)
• sum augij xj
• xi augiN - sum

27
Common Programming Errors

• Use constants for each dimensions size when
  declaring multidimensional array
• When declaring the array as a parameter of a
  function if you omit the first dimension all
  other dimensions must be supplied
• Since access to the elements of a
  multidimensional array requires nested counting
  loops it is easy to make out-of-range error.
• Since using multidimensional arrays as local
  variables requires large memory space, you may
  need to tell to operating system to increase
  stack size when the program is running