The document discusses addressing formulas for sparse matrices using column major representations in 1D arrays. It begins by introducing sparse matrices like lower triangular matrices and tridiagonal matrices. It then presents the row major representation for a lower triangular matrix, showing how to calculate the address of each element in the 1D array. Next, it proposes a column major representation for lower triangular matrices and tridiagonal matrices. Formulas are derived to calculate the address of each element in the column major 1D arrays. The column major approach stores elements column-wise rather than row-wise.
The document discusses 2D arrays, including their definition as an array of arrays, implementation methods of row-major and column-major ordering, and formulas for calculating element addresses based on storage method. It also covers basic arithmetic operations on 2D arrays like addition, subtraction, and multiplication. Examples of problems involving 2D arrays and their solutions are provided.
data structure and algorithm Array.pptx btech 2nd yearpalhimanshi999
The document discusses arrays and algorithms. It begins by defining arrays as lists of homogeneous data elements referenced by indices. Linear arrays are one-dimensional while multi-dimensional arrays include matrices. Common array operations like traversal, insertion, deletion and searching algorithms like linear and binary search are described. For linear search, the best, average and worst case complexities are analyzed to be O(1), O(n/2) and O(n) respectively. Binary search is more efficient, having a complexity of O(log n) as it halves the search space on each comparison. Examples and pseudocode are provided to illustrate key concepts.
Dr. Pradyumansinh Jadeja of Darshan Institute of Engineering & Technology discusses arrays and their applications. He explains one-dimensional and two-dimensional arrays, including how their elements are stored and addressed in memory. Applications of arrays include representing polynomial equations with matrices and storing sparse matrices efficiently through a linear representation that lists only non-zero elements.
All mathematical truths are relative and conditional. Determinants are numbers associated with square matrices that represent the signed area or volume of geometric objects defined by the matrices. This document defines properties of determinants, such as how they are calculated for different sized matrices, how operations on the rows and columns of a matrix affect its determinant, and properties like determinants of diagonal, symmetric, and skew-symmetric matrices. It also provides examples of using determinants to find equations of lines and the area of triangles.
This document discusses data structures and algorithms. It begins by classifying data structures as linear or non-linear, with linear data structures having each element connected to a unique predecessor and successor. It then discusses the two basic representations of linear data structures in memory as either arrays or linked lists. Common operations on linear data structures like traversal, search, insertion, deletion, sorting, and merging are also outlined. The document proceeds to provide detailed explanations and examples of arrays, including linear arrays, representation in memory, traversing, inserting, deleting, and multidimensional arrays. It concludes with a brief discussion of pointers and pointer arrays.
This document provides definitions and implementations of various operations on one-dimensional arrays in C++. It begins with definitions of a 1D array, including how to calculate its size and address of elements. It then discusses implementations of basic operations like insertion, deletion, searching and sorting. Linear and binary search algorithms are presented for searching arrays. Selection, bubble, and insertion sort algorithms are covered for sorting. The document also discusses merging two sorted arrays. Examples of array problems and their solutions are provided at the end.
1. The document discusses various types and operations of matrices including transpose, similarity, inverse, and determinant of matrices.
2. It also discusses using matrices to solve systems of linear equations by finding the inverse of the coefficient matrix or calculating the determinant.
3. The key matrix concepts covered are matrix notation, types of matrices, matrix addition/subtraction, multiplication, and using matrices to represent and solve linear systems.
The document discusses 2D arrays, including their definition as an array of arrays, implementation methods of row-major and column-major ordering, and formulas for calculating element addresses based on storage method. It also covers basic arithmetic operations on 2D arrays like addition, subtraction, and multiplication. Examples of problems involving 2D arrays and their solutions are provided.
data structure and algorithm Array.pptx btech 2nd yearpalhimanshi999
The document discusses arrays and algorithms. It begins by defining arrays as lists of homogeneous data elements referenced by indices. Linear arrays are one-dimensional while multi-dimensional arrays include matrices. Common array operations like traversal, insertion, deletion and searching algorithms like linear and binary search are described. For linear search, the best, average and worst case complexities are analyzed to be O(1), O(n/2) and O(n) respectively. Binary search is more efficient, having a complexity of O(log n) as it halves the search space on each comparison. Examples and pseudocode are provided to illustrate key concepts.
Dr. Pradyumansinh Jadeja of Darshan Institute of Engineering & Technology discusses arrays and their applications. He explains one-dimensional and two-dimensional arrays, including how their elements are stored and addressed in memory. Applications of arrays include representing polynomial equations with matrices and storing sparse matrices efficiently through a linear representation that lists only non-zero elements.
All mathematical truths are relative and conditional. Determinants are numbers associated with square matrices that represent the signed area or volume of geometric objects defined by the matrices. This document defines properties of determinants, such as how they are calculated for different sized matrices, how operations on the rows and columns of a matrix affect its determinant, and properties like determinants of diagonal, symmetric, and skew-symmetric matrices. It also provides examples of using determinants to find equations of lines and the area of triangles.
This document discusses data structures and algorithms. It begins by classifying data structures as linear or non-linear, with linear data structures having each element connected to a unique predecessor and successor. It then discusses the two basic representations of linear data structures in memory as either arrays or linked lists. Common operations on linear data structures like traversal, search, insertion, deletion, sorting, and merging are also outlined. The document proceeds to provide detailed explanations and examples of arrays, including linear arrays, representation in memory, traversing, inserting, deleting, and multidimensional arrays. It concludes with a brief discussion of pointers and pointer arrays.
This document provides definitions and implementations of various operations on one-dimensional arrays in C++. It begins with definitions of a 1D array, including how to calculate its size and address of elements. It then discusses implementations of basic operations like insertion, deletion, searching and sorting. Linear and binary search algorithms are presented for searching arrays. Selection, bubble, and insertion sort algorithms are covered for sorting. The document also discusses merging two sorted arrays. Examples of array problems and their solutions are provided at the end.
1. The document discusses various types and operations of matrices including transpose, similarity, inverse, and determinant of matrices.
2. It also discusses using matrices to solve systems of linear equations by finding the inverse of the coefficient matrix or calculating the determinant.
3. The key matrix concepts covered are matrix notation, types of matrices, matrix addition/subtraction, multiplication, and using matrices to represent and solve linear systems.
Some types of matrices, Eigen value , Eigen vector, Cayley- Hamilton Theorem & applications, Properties of Eigen values, Orthogonal matrix , Pairwise orthogonal, orthogonal transformation of symmetric matrix, denationalization of a matrix by orthogonal transformation (or) orthogonal deduction, Quadratic form and Canonical form , conversion from Quadratic to Canonical form, Order, Index Signature, Nature of canonical form.
This document discusses linear arrays and algorithms for common operations on arrays such as traversing, inserting, deleting, and sorting elements. It defines a linear array as a list of homogeneous data elements with a lower and upper bound. Algorithms provided include traversing from the lower to upper bound, inserting an element at a specified position by shifting other elements and incrementing the size, deleting an element by shifting elements and decrementing the size, and sorting an array using bubble sort by making multiple passes to swap adjacent elements in descending order.
The document discusses several advanced intermediate code generation techniques used in compilers, including:
1) Reusing temporary variable names to avoid using new temps whenever possible.
2) Addressing elements in one-dimensional and multi-dimensional arrays using base addresses and calculating offsets.
3) Translating logical, relational, and Boolean expressions using three-address code with temporary variables to store intermediate values.
The document discusses 2D arrays including their definition, implementation, and how to calculate the address of elements. It covers storing arrays in row-major and column-major order and includes formulas to calculate addresses based on the lower and upper bounds. Operations on 2D arrays like addition, subtraction, and multiplication are also explained. Some example problems are provided at the end to demonstrate calculating addresses of elements in 2D arrays stored in both row-major and column-major order.
Hermite integrators and 2-parameter subgroup of Riordan groupKeigo Nitadori
1. The document presents a generalization of 2-step Hermite integrators using a 2-parameter matrix M(a,b) whose elements are powers of a and b.
2. M(a,b) can be factorized into a lower triangular matrix L(a,b) and an upper triangular Pascal matrix Upas.
3. Explicit forms are derived for the inverse matrices M−1(a,b) which are needed for implementing higher order Hermite integrators.
Arrays are data structures that store a collection of data values of the same type in consecutive memory locations that can be individually referenced by their numeric index. Different representations of arrays are possible including using a single block of memory or a collection of elements. Common operations on arrays include retrieving or storing elements by their index and traversing the entire array sequentially.
BCA DATA STRUCTURES LINEAR ARRAYS MRS.SOWMYA JYOTHISowmya Jyothi
1) The document discusses linear arrays and their representation in memory. A linear array stores elements in consecutive memory locations.
2) The number of elements in an array is given by the length formula: Length = Upper Bound - Lower Bound + 1. The address of each element can be calculated using the base address of the array.
3) Common array operations like traversing, inserting, and deleting elements are discussed along with algorithms to perform each operation. Representing polynomials and sparse matrices using arrays is also covered.
The document discusses determinants and how to calculate them. It defines a determinant as a pure number associated with a square matrix. It provides an example of calculating the determinant of a 2x2 matrix. It then explains how to calculate minors and cofactors, which are used in Cramer's rule to solve systems of linear equations by calculating the determinants of the original and augmented matrices.
This document provides instructions for an assignment on data and file structures lab. It includes two questions to write programs in C language on:
1. Writing an algorithm and program to convert a tree to a binary tree.
2. Writing an algorithm and program for multiplication of two sparse matrices using pointers.
The document provides detailed algorithms and C code implementations for converting a binary tree to a binary search tree and representing and multiplying sparse matrices using multilinked data structures with pointers.
The document describes several algorithms:
1) Algorithm 5.1.4 tiles a deficient n x n board with trominoes by recursively dividing the board into quarters and placing a tromino in the center.
2) Algorithm 5.2.2 merges two sorted subarrays into a single sorted array.
3) Algorithm 5.2.3 (Mergesort) sorts an array by recursively dividing it in half and merging the sorted halves.
4) Algorithm 5.3.2 finds the closest pair of points in an array of points by recursively dividing the points into halves sorted by x-coordinate and merging while maintaining the distance between closest points.
The document describes several algorithms:
1) Algorithm 5.1.4 tiles a deficient n x n board with trominoes by recursively dividing the board into quarters and placing a tromino in the center.
2) Algorithm 5.2.2 merges two sorted subarrays into a single sorted array.
3) Algorithm 5.2.3 (Mergesort) sorts an array by recursively dividing it in half and merging the sorted halves.
4) Algorithm 5.3.2 finds the closest pair of points in an array of points by recursively dividing the points into halves sorted by x-coordinate and merging while maintaining the distance between closest points.
The document discusses the merge sort algorithm. It works by recursively dividing an array into two halves, sorting each half, and then merging the sorted halves back together. The key steps are:
1) Divide the array into equal halves recursively until arrays contain a single element.
2) Sort the halves by recursively applying the merge sort algorithm.
3) Merge the sorted halves back into a single sorted array by comparing elements and copying the smaller value into the output array.
The document discusses divide and conquer algorithms. It explains that divide and conquer algorithms follow three steps: 1) divide the problem into subproblems, 2) conquer the subproblems by solving them recursively, and 3) combine the results to solve the original problem. Binary search, merge sort, and quicksort are provided as examples of divide and conquer algorithms. Binary search divides a sorted array in half at each step to search for a target value. Merge sort divides the array in half, recursively sorts the halves, and then merges the sorted halves. Quicksort chooses a pivot to partition the array into left and right halves, recursively sorts the halves, and returns the fully sorted array.
The document discusses arrays in C programming. It defines arrays as a collection of similar data types stored in contiguous memory locations and accessed using indexes. It describes how to declare, initialize, access, and manipulate single and multi-dimensional arrays. It also covers passing arrays to functions, array addressing and calculations, and merging two sorted arrays.
The document discusses various sorting algorithms including insertion sort, selection sort, bubble sort, merge sort, and quick sort. It provides detailed explanations of how each algorithm works through examples using arrays or lists of numbers. The key steps of each algorithm are outlined in pseudocode to demonstrate how they sort a set of data in either ascending or descending order.
The document discusses arrays in C programming. It defines key concepts like array representation, index starting from 0, memory representation of arrays, and address calculation of array elements. It also covers various array operations like creation, copying, deletion, insertion, sorting, and representation of arrays as abstract data types. Multidimensional arrays and their row-major and column-major representations are explained along with examples.
Matrices can be added or subtracted if they are the same size. To multiply matrices, each element in a row of the first matrix is multiplied by the corresponding element in a column of the second matrix and those products are summed. The result is a matrix where the number of rows equals the rows of the first matrix and the number of columns equals the columns of the second matrix. There are different types of matrices including square, rectangular, diagonal, identity, and null matrices. Matrix operations allow the representation and solution of systems of equations.
This document provides an overview of mathematical functions and plotting in MATLAB. It discusses:
- The many predefined mathematical functions available in MATLAB, such as sin, cos, and exp.
- How to generate vectors and matrices for input.
- How to perform basic plotting of data and functions using commands like plot and linspace.
- How to customize plots by adding titles, labels, and changing colors.
- Matrix operations in MATLAB like transposes, inverses, concatenation and arithmetic operations.
2. Linear Data Structure Using Arrays - Data Structures using C++ by Varsha P...widespreadpromotion
This document discusses arrays and ordered lists as linear data structures. It describes arrays as a collection of homogeneous data elements that allow direct access via indexes. The key properties of arrays include sequential storage of elements, random access, and calculation of element addresses based on base address and offsets. The document also covers array implementation in C++, one-dimensional and two-dimensional arrays, row-major and column-major representations, and the use of arrays to represent polynomials, strings and sparse matrices. Finally, it provides a brief introduction to ordered lists and their representation as linear data structures.
ACEP Magazine edition 4th launched on 05.06.2024Rahul
This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
Electric vehicle and photovoltaic advanced roles in enhancing the financial p...IJECEIAES
Climate change's impact on the planet forced the United Nations and governments to promote green energies and electric transportation. The deployments of photovoltaic (PV) and electric vehicle (EV) systems gained stronger momentum due to their numerous advantages over fossil fuel types. The advantages go beyond sustainability to reach financial support and stability. The work in this paper introduces the hybrid system between PV and EV to support industrial and commercial plants. This paper covers the theoretical framework of the proposed hybrid system including the required equation to complete the cost analysis when PV and EV are present. In addition, the proposed design diagram which sets the priorities and requirements of the system is presented. The proposed approach allows setup to advance their power stability, especially during power outages. The presented information supports researchers and plant owners to complete the necessary analysis while promoting the deployment of clean energy. The result of a case study that represents a dairy milk farmer supports the theoretical works and highlights its advanced benefits to existing plants. The short return on investment of the proposed approach supports the paper's novelty approach for the sustainable electrical system. In addition, the proposed system allows for an isolated power setup without the need for a transmission line which enhances the safety of the electrical network
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Some types of matrices, Eigen value , Eigen vector, Cayley- Hamilton Theorem & applications, Properties of Eigen values, Orthogonal matrix , Pairwise orthogonal, orthogonal transformation of symmetric matrix, denationalization of a matrix by orthogonal transformation (or) orthogonal deduction, Quadratic form and Canonical form , conversion from Quadratic to Canonical form, Order, Index Signature, Nature of canonical form.
This document discusses linear arrays and algorithms for common operations on arrays such as traversing, inserting, deleting, and sorting elements. It defines a linear array as a list of homogeneous data elements with a lower and upper bound. Algorithms provided include traversing from the lower to upper bound, inserting an element at a specified position by shifting other elements and incrementing the size, deleting an element by shifting elements and decrementing the size, and sorting an array using bubble sort by making multiple passes to swap adjacent elements in descending order.
The document discusses several advanced intermediate code generation techniques used in compilers, including:
1) Reusing temporary variable names to avoid using new temps whenever possible.
2) Addressing elements in one-dimensional and multi-dimensional arrays using base addresses and calculating offsets.
3) Translating logical, relational, and Boolean expressions using three-address code with temporary variables to store intermediate values.
The document discusses 2D arrays including their definition, implementation, and how to calculate the address of elements. It covers storing arrays in row-major and column-major order and includes formulas to calculate addresses based on the lower and upper bounds. Operations on 2D arrays like addition, subtraction, and multiplication are also explained. Some example problems are provided at the end to demonstrate calculating addresses of elements in 2D arrays stored in both row-major and column-major order.
Hermite integrators and 2-parameter subgroup of Riordan groupKeigo Nitadori
1. The document presents a generalization of 2-step Hermite integrators using a 2-parameter matrix M(a,b) whose elements are powers of a and b.
2. M(a,b) can be factorized into a lower triangular matrix L(a,b) and an upper triangular Pascal matrix Upas.
3. Explicit forms are derived for the inverse matrices M−1(a,b) which are needed for implementing higher order Hermite integrators.
Arrays are data structures that store a collection of data values of the same type in consecutive memory locations that can be individually referenced by their numeric index. Different representations of arrays are possible including using a single block of memory or a collection of elements. Common operations on arrays include retrieving or storing elements by their index and traversing the entire array sequentially.
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1) The document discusses linear arrays and their representation in memory. A linear array stores elements in consecutive memory locations.
2) The number of elements in an array is given by the length formula: Length = Upper Bound - Lower Bound + 1. The address of each element can be calculated using the base address of the array.
3) Common array operations like traversing, inserting, and deleting elements are discussed along with algorithms to perform each operation. Representing polynomials and sparse matrices using arrays is also covered.
The document discusses determinants and how to calculate them. It defines a determinant as a pure number associated with a square matrix. It provides an example of calculating the determinant of a 2x2 matrix. It then explains how to calculate minors and cofactors, which are used in Cramer's rule to solve systems of linear equations by calculating the determinants of the original and augmented matrices.
This document provides instructions for an assignment on data and file structures lab. It includes two questions to write programs in C language on:
1. Writing an algorithm and program to convert a tree to a binary tree.
2. Writing an algorithm and program for multiplication of two sparse matrices using pointers.
The document provides detailed algorithms and C code implementations for converting a binary tree to a binary search tree and representing and multiplying sparse matrices using multilinked data structures with pointers.
The document describes several algorithms:
1) Algorithm 5.1.4 tiles a deficient n x n board with trominoes by recursively dividing the board into quarters and placing a tromino in the center.
2) Algorithm 5.2.2 merges two sorted subarrays into a single sorted array.
3) Algorithm 5.2.3 (Mergesort) sorts an array by recursively dividing it in half and merging the sorted halves.
4) Algorithm 5.3.2 finds the closest pair of points in an array of points by recursively dividing the points into halves sorted by x-coordinate and merging while maintaining the distance between closest points.
The document describes several algorithms:
1) Algorithm 5.1.4 tiles a deficient n x n board with trominoes by recursively dividing the board into quarters and placing a tromino in the center.
2) Algorithm 5.2.2 merges two sorted subarrays into a single sorted array.
3) Algorithm 5.2.3 (Mergesort) sorts an array by recursively dividing it in half and merging the sorted halves.
4) Algorithm 5.3.2 finds the closest pair of points in an array of points by recursively dividing the points into halves sorted by x-coordinate and merging while maintaining the distance between closest points.
The document discusses the merge sort algorithm. It works by recursively dividing an array into two halves, sorting each half, and then merging the sorted halves back together. The key steps are:
1) Divide the array into equal halves recursively until arrays contain a single element.
2) Sort the halves by recursively applying the merge sort algorithm.
3) Merge the sorted halves back into a single sorted array by comparing elements and copying the smaller value into the output array.
The document discusses divide and conquer algorithms. It explains that divide and conquer algorithms follow three steps: 1) divide the problem into subproblems, 2) conquer the subproblems by solving them recursively, and 3) combine the results to solve the original problem. Binary search, merge sort, and quicksort are provided as examples of divide and conquer algorithms. Binary search divides a sorted array in half at each step to search for a target value. Merge sort divides the array in half, recursively sorts the halves, and then merges the sorted halves. Quicksort chooses a pivot to partition the array into left and right halves, recursively sorts the halves, and returns the fully sorted array.
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The document discusses various sorting algorithms including insertion sort, selection sort, bubble sort, merge sort, and quick sort. It provides detailed explanations of how each algorithm works through examples using arrays or lists of numbers. The key steps of each algorithm are outlined in pseudocode to demonstrate how they sort a set of data in either ascending or descending order.
The document discusses arrays in C programming. It defines key concepts like array representation, index starting from 0, memory representation of arrays, and address calculation of array elements. It also covers various array operations like creation, copying, deletion, insertion, sorting, and representation of arrays as abstract data types. Multidimensional arrays and their row-major and column-major representations are explained along with examples.
Matrices can be added or subtracted if they are the same size. To multiply matrices, each element in a row of the first matrix is multiplied by the corresponding element in a column of the second matrix and those products are summed. The result is a matrix where the number of rows equals the rows of the first matrix and the number of columns equals the columns of the second matrix. There are different types of matrices including square, rectangular, diagonal, identity, and null matrices. Matrix operations allow the representation and solution of systems of equations.
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- How to generate vectors and matrices for input.
- How to perform basic plotting of data and functions using commands like plot and linspace.
- How to customize plots by adding titles, labels, and changing colors.
- Matrix operations in MATLAB like transposes, inverses, concatenation and arithmetic operations.
2. Linear Data Structure Using Arrays - Data Structures using C++ by Varsha P...widespreadpromotion
This document discusses arrays and ordered lists as linear data structures. It describes arrays as a collection of homogeneous data elements that allow direct access via indexes. The key properties of arrays include sequential storage of elements, random access, and calculation of element addresses based on base address and offsets. The document also covers array implementation in C++, one-dimensional and two-dimensional arrays, row-major and column-major representations, and the use of arrays to represent polynomials, strings and sparse matrices. Finally, it provides a brief introduction to ordered lists and their representation as linear data structures.
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Electric vehicle and photovoltaic advanced roles in enhancing the financial p...IJECEIAES
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2. Introduction
Sparse Matrix
Lower Triangular Matrix
Tri-diagonal Matrix
Row Major Representation For Lower Triangular Matrix
Column Major Representation For Lower Triangular Matrix
Column Major Representation For Tri-diagonal Matrix
Advantages of Proposed Method
References
TABLE OF CONTENTS
2
3. INTRODUCTION
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Sparse matrix generally contains elements with zero values; examples of such kind of matrices are lower & upper
triangular matrix.
For the solution of linear equations, these matrices have been used .
Since many of the elements in such matrices contain zeroes. Thus a need to save space arises, either in memory or
disk.
For lower and upper triangular matrices of 1-D array, analysis has been done. After the analysis their
corresponding addressing formulas have been developed.
4. Few variants are also analyzed for lower and upper triangular matrices.
It is needed to mention that different approaches for addressing an element of sparse matrices are remain
unanalyzed in which column major are also included.
In this paper we analyze storage by means of 1-D array for sparse matrices of the lower triangular type, plus one
matrix of the tri-diagonal type.
To address an element in the lower triangular matrices using row major is presented first.
After this, a new approach based on column major is defined and analyzed for lower triangular matrices and tri-
diagonal matrices.
INTRODUCTION…
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5. SPARSE MATRIX
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Matrices with a relatively high proportion of zero entries are called sparse matrices.
Two general types of N-square sparse matrices, which occur in various applications, are
Lower Triangular Matrix (LTM)
Tri-diagonal Matrix (TDM)
6. A matrix which contains nonzero elements only on or below the main
diagonal is called a lower triangular matrix i.e. above the main
diagonal all elements are zero.
Let us define a Lower Triangular Matrix (LTM) named A of order N,
N > 0. Fig. 1 shows an example of this kind of matrix, with N = 7
There are 7 rows and 7 columns in above 7-square LTM A. Any
element in LTM A is represented by a(J, K) where
J=0, 1, 2….7 and K=0, 1, 2….7
In LTM A, row-1 contains 1 element and row-2 contains 2 elements.
Similarly row-3, row-4….row-7 contains 3, 4….7 numbers of
elements respectively. From above Fig. 1, it may be observed that
total number of elements in any LTM of order N can be calculated by
arithmetic progression as follows:
1+2+3+…. +N = N(N+1)/2 (1)
Hence in LTM A, there are 7*(7+1)/2 = 28 elements.
LOWER TRIANGULAR MATRIX(LTM)
Fig. 1. Lower Triangular Matrix A (LTM A)
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1 2 3 4 5 6 7
1 4
2 3 -5
3 1 0 6
4 -7 8 -1 3
5 5 -2 0 2 -2
6 8 4 5 -1 6 -8
7 2 7 9 0 9 5 -4
7. The matrix, where nonzero entries can only occur on the diagonal or
on elements immediately above or below the diagonal, is called a tri-
diagonal matrix.
Let us define a Tri-diagonal Matrix (TDM) named B of order N, N >
0. Fig. 2 shows an example of this kind of matrix, with N = 7
There are 7 rows and 7 columns in above 7-square TDM B. Any
element in TDM B is represented by b(J, K) where
J=0, 1, 2….7 and K=0, 1, 2….7
Note that TDM has N=7 elements on the diagonal and N–1=6
elements above and below the diagonal. From above Fig. 2, it may be
observed that total number of elements in any TDM of order N can
be calculated as follows:
N + (N–1) + (N–1) = 3N – 2 (2)
Hence TDM B contains at most 3N–2 nonzero elements i.e.
3*7 – 2 = 19.
TRI-DIAGONAL MATRIX(TDM)
Fig. 2. Tri-diagonal Matrix B (TDM B)
7
1 2 3 4 5 6 7
1 4 7
2 3 -5 4
3 0 6 1
4 -1 3 -3
5 2 -2 0
6 6 -8 2
7 5 -4
8. ROW MAJOR REPRESENTATION FOR LTM
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4 3 -5 1 0 6 -7 8 -1 3 5 -2 0 2 -2 8 4 5 -1 6 -8 2 7 9 0 9 5 -4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Fig. 3. 1-D representation by rows of LTM A
Elements in LTM A are represented in Fig. 3 using 1-D array named C.
Any element in 1-D array C is represented by C[L] where L=0, 1, 2….28
Given an element a(J, K) belonging to the triangular array LTM A. Here, the purpose is to determine the
corresponding position of same element a(J, K) in 1-D array C.
The position of element a(J, K) in array C is determined in the following way.
C[1]=a(1,1), C[2]=a(2,1).…C[L]=a(J,K)….C[28]=a(7,7)
According to equation (1), number of elements in any LTM of order N is N(N+1)/2.
So it may be observed that 1-D array C contains only N(N+1)/2 number of elements i.e. 28.
9. ROW MAJOR REPRESENTATION FOR LTM…
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Since the position of a(J, K) is required in 1-D array C. So the purpose is to calculate the formula that
gives us the integer L in terms of J and K where C[L] = a(J, K).
I.e. L gives the address of element a(J, K) in which all the elements including a(J, K) are list up in 1-D
array C.
To calculate the address of element a(J, K) in 1-D array C, following algorithm is used.
Step-1: Calculate the number of elements coming before Jth row, i.e. number of elements up to (J-1)th row.
Step-2: Then calculate the number of elements in Jth row including a(J, K).
Step-3: Add both the values calculated in above two steps. The result is desired value L i.e. address of element
a(J, K) in 1-D array C.
10. Total numbers of elements in the rows above the element a(J, K) are calculated by arithmetic progression as
follows:
1+2+3+….+(J–1) = J(J–1)/2 (3)
and there are K elements in row-J up to and including a(J, K). The address of element a(J, K) can be
calculated by adding equation (3) with number of elements in row-J up to and including a(J, K) i.e. K. So
L = J(J–1)/2 + K (4)
yields the index that accesses the value of element a(J, K) from the linear array C.
For example: Let’s calculate the address of an element a(5, 3) of LTM A in array C using above formula.
Here J = 5 and K = 3 Then
L = 5(5–1)/2 + 3
L= 5*4/2 + 3
L = 10 + 3
L=13
It means the address of the desired element in 1-D array C is 13. It may be observed in Fig. 3.
ROW MAJOR REPRESENTATION FOR LTM…
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The approach is to insert the same elements of LTM A in another 1-D array named D column wise.
Elements in LTM A are represented in Fig. 4 using 1-D array D.
Any element in 1-D array D is represented by D[L] where L=0, 1, 2….28
Given an element a(J, K) belonging to the triangular array LTM A. Here, the purpose is to determine the
corresponding position of same element a(J, K) in 1-D array D.
The position of element a(J, K) in array D is determined in the following way.
D[1]=a(1,1), D[2]=a(2,1), D[3]=a(3,1) .…D[L]=a(J,K)…. D[28]=a(7,7)
COLUMN MAJOR REPRESENTATION FOR LTM
4 3 1 -7 5 8 2 -5 0 8 -2 4 7 6 -1 0 5 9 3 2 -1 0 -2 6 9 -8 5 -4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Fig. 4. 1-D representation by columns of LTM A
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Since the position of a(J, K) is required in 1-D array D. So the purpose is to calculate the formula that gives us the
integer L in terms of J and K where D[L] = a(J, K).
I.e. L gives the address of element a(J, K) in which all the elements including a(J, K) are list up in 1-D array D.
To calculate the address of element a(J, K) in 1-D array D, following algorithm is used.
Step-1: Calculate the number of elements coming after Kth column, i.e. total number of elements coming
from (K+1)th column to Nth column.
Step-2: Then calculate the number of elements in Kth column coming after the element a(J, K).
Step-3: Add both the values calculated in above two steps. The result is total number of elements coming
after element a(J, K).
Step-4: Subtract the value of Step-3 from equation (1) i.e. total number of elements in LTM A. The result is
desired value L i.e. address of element a(J, K) in 1-D array D.
COLUMN MAJOR REPRESENTATION FOR LTM…
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Total number of elements coming from (N–K)th i.e. (K+1)th to Nth column can be calculated by arithmetic
progression as follows:
1 + 2 + 3 + …. + (N–K) = (N–K)(N–K+1)/2 (5)
and there are (N-J) elements in same column Kth after a(J, K).
So total number of elements after element a(J, K) are:
{(N–K)(N–K+1)/2} + (N–J) (6)
According to Step-4, address of element a(J, K) of LTM A can be calculated by subtracting the total number of
elements coming after a(J, K) from total number of elements in LTM A. Subtract equation (6) from equation (1)
i.e.
L= {N(N + 1)/2} – {(N–K)(N–K+1)/2 + (N–J)} (7)
yields the index that accesses the value of element a(J, K) from the linear array D.
For example: calculate the address of same element a(5, 3) of LTM A in array D, according to proposed method.
Here J = 5 , K = 3 and N = 7
Then L = {7(7+1)/2} – {(7–3)(7–3+1)/2 + (7–5)}
L = 28 – {(4*5/2) + 2}
L = 28 – 12 = 16
It means the address of the same element in 1-D array D is 16. The same may be observed in Fig. 4.
COLUMN MAJOR REPRESENTATION FOR LTM…
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The approach is to insert the elements of TDM B in another 1-D array named E column wise. Elements in TDM B
are represented in Fig. 5 using 1-D array E.
Any element in 1-D array E is represented by E[L] where L=0, 1, 2….19
Given an element b(J, K) belonging to the TDM B. Here, the purpose is to determine the corresponding position of
same element b(J, K) in 1-D array E.
The position of the element b(J, K) in array E is determined in the following way.
E[1]=b(1,1), E[2]=b(2,1), E[3]=b(1,2) .…E[L]=b(J,K)…. E[19]=b(7,7)
It may be observed initially from equation (2) that array E contains only (3N – 2) number of elements i.e. 19.
COLUMN MAJOR REPRESENTATION FOR TDM
4 3 7 -5 0 4 6 -1 1 3 2 -3 -2 6 0 -8 5 2 -4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Fig. 5. 1-D representation by columns of TDM B
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Since the position of b(J, K) is required in 1-D array E. So the purpose is to calculate the formula that gives us the
integer L in terms of J and K where E[L] = b(J, K).
I.e. L gives the address of element b(J, K) in which all the elements including b(J, K) are list up in 1-D array E.
To calculate the address of element b(J, K) in 1-D array E, following algorithm is used.
Step-1: Calculate the number of elements coming before Kth column, i.e. number of elements upto (K–1)th
column.
Step-2: Then calculate the number of elements in Kth column coming before element b(J, K).
Step-3: Add both the values calculated in above two steps and also add 1 in it for b(J, K) itself. The result is
desired value L i.e. address of element b(J, K) in 1-D array E.
COLUMN MAJOR REPRESENTATION FOR TDM…
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Total number of elements coming before the column of element b(J, K) is
3(K–2) + 2 (8)
Total number of elements coming before b(J, K) in Kth column is (J–K+1).
According to Step-3, address of element b(J, K) in 1-D array E can be calculated by adding above value (J–K+1)
and equation (8). Further add 1 for b(J, K) itself. So
L= [3(K–2)+2] + [J–K+1] + 1
L= 2K + J – 2 (9)
yields the index that accesses the value of element b(J, K) from the linear array E.
For example, Let’s calculate the address of an element b(5, 6) of TDM B in array E, according to proposed
method.
Here J = 5, K = 6 and N = 7
Then
L = 2*6 + 5 – 2
L = 12 + 5 – 2
L = 15
It means the address of the element in 1-D array E is 15. It can be observed in Fig. 5.
COLUMN MAJOR REPRESENTATION FOR TDM…
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Two methods are generally used to traverse the elements of any sparse matrix.
The first method is row wise and second method is column wise.
During traversal few of the elements appears first in column wise rather than row wise.
In this scenario proposed method is useful to access the elements easily.
ADVANTAGES OF PROPOSED METHOD
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REFERENCES
[1] Grossman S., “Álgebra Lineal,” McGraw Hill, 1996.
[2] Cairó O. & Guardati S., “Data Structures,” McGraw−Hill, 2005.
[3] Horowitz E. & Sahni S., “Fundamentals of data Structures in Pascal,” Computer Science Press, 1990.
[4] Moreno F. & Flórez R., “Fórmulas de Direccionamiento en Matrices Triangulares,” Journal Facultad de Ingeniería
Universidad de Antioquia. Medellín, 2001.
[5] Seymour Lipschutz, “Data Structures,” McGraw-Hill, 2014.