(Creating Arrays & Matrices) Applied Linear Algebra in Geoscience Using MATLAB

Transcription

1 Applied Linear Algebra in Geoscience Using MATLAB (Creating Arrays & Matrices)

2 Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in MATLAB User-Defined Functions and Function Files Polynomials, Curve Fitting, and Interpolation Applications in Numerical Analysis Three-Dimensional Plots Symbolic Math Matrices Linear equations Determinants Eigenvalues and eigenvectors Orthogonal vectors and matrices Vector and matrix norms Gaussian elimination and the LU dec. Linear system applications Gram-Schmidt decomposition The singular value decomposition Least-squares problems Implementing the QR factorization The algebraic eigenvalue problem

3 Creating a 1D array (Vector) A one-dimensional array is a list of numbers arranged in a row or a column. The vector is created by typing the elements (numbers) inside square brackets [ ]. The position of point A can be expressed in terms of a position vector: Note: A vector can also be created as the result of mathematical operations

4 Creating a 1D array (Vector) Creating a vector with constant spacing If only two numbers (the first and the last terms) are typed (the spacing is omitted), then the default for the spacing is 1.

5 Creating a 1D array (Vector) Creating a vector with linear (equal) spacing by specifying the number of terms

6 Creating a 2D array (Matrix) A two-dimensional array, also called a matrix, has numbers in rows and columns. A matrix is created by assigning the elements of the matrix to a variable. This is done by typing the elements, row by row, inside square brackets [ ]. Rows of a matrix can also be entered as vectors using the notation for creating vectors with constant spacing, or the linspace command. For example:

7 Creating a 2D array (Matrix) The zeros(m,n), ones(m,n), and eye(n) commands can be used to create matrices that have elements with special values All variables in MATLAB are arrays. A scalar is an array with one element, a vector is an array with one row or one column of elements, and a matrix is an array with elements in rows and columns.

8 The Transpose Operator The transpose operator, when applied to a vector, switches a row (column) vector to a column (row) vector. When applied to a matrix, it switches the rows (columns) to columns (rows). The transpose operator is applied by typing a single quote following the variable to be transposed. Examples are:

9 Array Addressing (Vector) The address of an element in a vector is its position in the row (or column).

10 Array Addressing (Matrix) The address of an element in a matrix is its position, defined by the row number and the column number where it is located.

11 Using A Colon (:) A colon can be used to address a range of elements in a vector or a matrix. For a vector: va(:) Refers to all the elements of the vector va (either a row or a column vector). va(m:n) Refers to elements m through n of the vector va. For a matrix: A(:,n) Refers to the elements in all the rows of column n of the matrix A. A(n,:) Refers to the elements in all the columns of row n of the matrix A. A(:,m:n) Refers to the elements in all the rows between columns m and n of the matrix A. A(m:n,:) Refers to the elements in all the columns between rows m and n of the matrix A. A(m:n,p:q) Refers to the elements in rows m through n and columns p through q of the matrix A.

12 Using A Colon (:) It is possible, to select only specific elements, or specific rows and columns of existing variables to create new variables. This is done by typing the selected elements or rows or columns inside brackets, as shown ==>

13 Adding Elements Adding elements to a vector: Elements can be added to an existing vector by assigning values to the new elements.

14 Adding Elements Adding elements to a matrix: The size of the added rows or columns must fit the existing matrix!

15 Adding Elements If a matrix has a size of m,n and a new value is assigned to an element with an address beyond the size of the matrix, MATLAB increases the size of the matrix to include the new element. Zeros are assigned to the other elements that are added. Examples:

16 Deleting Elements An element, or a range of elements, of an existing variable can be deleted by reassigning Nothing to these elements. This is done by using square brackets with nothing typed in between them

17 Handling Arrays Functions MATLAB has many built-in functions for managing and handling arrays. Some of these are listed here:

18 Strings A string is an array of characters. It is created by typing the characters within single quotes. Strings can include letters, digits, other symbols, and spaces. Examples of strings: 'ad ef ', '3%fr2', '{edcba:21!', 'MATLAB'. A string that contains a single quote is created by typing two single quotes within the string. When a string is being typed in, the color of the text on the screen changes to maroon when the first single quote is typed. When the single quote at the end of the string is typed, the color of the string changes to purple. Strings can also be assigned to variables by simply typing the string on the right side of the assignment operator, as shown in the examples below

19 Strings MATLAB has a built-in function named char that creates an array with rows having the same number of characters from an input of rows not all of the same length. MATLAB makes the length of all the rows equal to that of the longest row by adding spaces at the end of the short lines. In the char function, the rows are entered as strings separated by a comma according to the following format: NOTE: