CPS 125: Digital Computation and Programming

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CPS 125 Digital Computation and Programming

• Multidimensional Arrays

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Outline

• Declaring and Referencing Multidimensional Arrays
• Modeling with Two-Dimensional Arrays
• Vectors and Matrices
• Solving Systems of Linear Equations
• Common Programming Errors

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Dimensions

• int num6

int num26
num00
num0
num1
num15
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Multidimensional Arrays

• Multiple dimensional arrays
• int room236
• Can think of as 2 sets of 3 rows and 6
  columns

room105
room000
room115
room125
room020
room025
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Multidimensional Array

• Really stored in contiguous fashion
• int room236

000
010
020
125
100
110
120
Just convenient to think of as 2 sets of 3
rows and 6 columns!
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Multidimensional Arrays

• Declaration initialization
• Similar to one-dimensional
• Nested braces, commas
• char name38 S, a, m, \0,
• C, h, u, c, k,
  \0,
• M, a, r, k, \0
• --- or ---
• char name38 Sam, Chuck, Mark
• int num26 1, 2, 3, 4, 5, 6, 2, 3

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Passing Multidimensional Arrays

• All but 1st subscript must be present
• function call init_array(rooms)
• function header void init_array(int array
  36)

Highest dimension always unspecified because we
may pass arrays of different sizes. Other
dimensions just say how array will be manipulated.
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• / 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)

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• Which of the arrays shown would be valid actual
  arguments to pass to a function as below,
• int fun(double arr310)
• double m6310
• int n4310
• double z8211
• double q20310
• How many elements are in each of arrays above?

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• int soil_type47MAXDEPTH
• /Finds and displays how much silt is contained
  in each north-south strip /
• for (north_south0 north_southlt4 north_south)
  
• silt_amount 0
• for (east_west0 east_westlt7 east_west)
• for (depth0 depthltMAXDEPTH depth)
• silt_amount soil_typenorth_southeast_west
  
• depth
• printf(There are d soil cells containing silt
  in north_south segment d.\n, silt_amount,
  north_south)

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Programming Project

• You have been asked to write one part of a
  detector analysis software package for a
  telescope. Your program takes as input the
  brightness of each point in a two-dimensional
  array representing an image of the sky. Use a
  1010 integer array for this image. Find and
  display the x and y coordinates and the value of
  the brightest pixel. If more than one pixel has
  this highest value, information for all
  highest-valued pixels should be displayed.

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Vectors

• Vector a mathematical object consisting of a
  sequence of numbers.
• / a vector lt4, 12, 19gt /
• int vect3 4, 12, 19
• Calculating scalar product
• sum_prod 0
• for (k0 kltn k)
• sum_prod xk wk

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Matrices

• Matrix a mathematical object consisting of a
  rectangular arrangement of numbers.
• / a matrix 4.1 8.3 /
• 7.9 6.2
• double m22 4.1, 8.3, 7.9, 6.2
• Multiplying a matrix by a vector
• vi 0
• for (k0 kltn k)
• vk aik xk

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• / 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

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• If m is a 5x5 integer matrix, what is displayed
  by this loop
• for (i0 i lt5 i)
• printf(8d, m2i)
• what is displayed by this loop?
• for (i0 i lt5 i)
• printf(8d, mi4)
• If sq is a C array representing the matrix
• 4 5 18
• 2 4 9
• 8 4 12
• what is displayed by this loop?
• for (i0 i lt3 i)
• printf(8d, sqii)

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Solving Systems of Linear Equations

• Algorithm for Gaussian Elimination
• Transform the original system into scaled
  triangular form
• Solve for the xi by back substitution
• Data structure augmented matrix
• Triangularizing and scaling rules
• Multiply any row of Aug by a nonzero number
• Add to any row of Aug a multiple of any other row
• Swap any two rows

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Solving Systems of Linear Equations

• Implementation
• Function Gauss
• Function pivot
• Function back substitution
• It works when there is unique solution
• It does not work for systems with no solution,
  multiple solutions, or ill-conditioned
• Computational round-off errors may be too large

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Common Programming Errors

• Must supply all except first dimensions for array
  parameters in function prototype
• Subscript range errors
• No enough space for large multidimensional arrays?