MATLAB Arrays - Computational Methods - Lecture Slides, Slides of Calculus for Engineers

Download MATLAB Arrays - Computational Methods - Lecture Slides and more Slides Calculus for Engineers in PDF only on Docsity! Engr/Math/Physics 25 Chp2 MATLAB Arrays: Part-1 Docsity.com Learning Goals • Learn to Construct 1-Dimensional Row and Column Vectors • Create MULTI-Dimensional ARRAYS and MATRICES • Perform Arithmetic Operations on Vectors and Arrays/Matrices • Analyze Polynomial Functions Docsity.com “Appending” Vectors • Can create a NEW vector by ''appending'' one vector to another • e.g.; to create the row vector u whose first three columns contain the values of r = [2,4,20] and whose 4th, 5th, & 6th columns contain the values of w = [9,-6,3] – Typing u = [r,w] yields vector u = [2,4,20,9,-6,3] >> r = [2,4,20]; w = [9,-6,3]; >> u = [r,w] u = 2 4 20 9 -6 3 Docsity.com Colon (:) Operator • The colon operator (:) easily generates a large vector of regularly spaced elements. • Typing >>x = [m:q:n] – creates a vector x of values with a spacing q. The first value is m. The last value is n IF m - n is an integer multiple of q. IF NOT, the last value is LESS than n. >> p = [0:2:8] p = 0 2 4 6 8 >> r = [0:2:7] r = 0 2 4 6 Docsity.com Colon (:) Operator cont. • To create a row vector z consisting of the values from 5 to 8 in steps of 0.1, type z = [5:0.1:8]. • If the increment q is OMITTED, it is taken as the DEFAULT Value of +1. >> s =[-11:-6] s = -11 -10 -9 -8 -7 -6 >> t = [-11:1:-6] t = -11 -10 -9 -8 -7 -6 Docsity.com logspace Example • Consider this command >> y =logspace(-1,2,6) y = 0.1000 0.3981 1.5849 6.3096 25.1189 100.0000  Calculate the Power of 10 increment ( ) ( )1−−=∆ nabp  In this case ( )( ) ( ) 6.053 1612 == −−−=∆p ( )( )110 −∆+= kpaky  In this Case k Power yk 1 -1 0.1000 2 -0.4 0.3981 3 0.2 1.5849 4 0.8 6.3096 5 1.4 25.1189 6 2 100.0000  for k = 1→6, then the kth y-value Docsity.com logspace Example (alternative) • Consider this command >> y =logspace(-1,2,6) y = 0.1000 0.3981 1.5849 6.3096 25.1189 100.0000  Take base-10 log of the above  Calc Spacing Between Adjacent Elements with diff command >> yLog10 = log10(y) yLog10 = -1.0000 -0.4000 0.2000 0.8000 1.4000 2.0000 >> yspc = diff(yLog10) yspc = 0.6000 0.6000 0.6000 0.6000 0.6000 ( ) ( ) [ ]( ) ( ) 5316121 =−−−=−−=∆ nabp Docsity.com Vector: Magnitude, Length, and Absolute Value • Keep in mind the precise meaning of these terms when using MATLAB – The length command gives the number of elements in the vector – The magnitude of a vector x having elements x1, x2, …, xn is a scalar, given by √(x12 + x22 + … + xn2), and is the same as the vector's geometric length – The absolute value of a vector x is, in MATLAB, a vector whose elements are the arithmetic absolute values of the elements of x Docsity.com 3D Vector Length Example - Run vAB = 2 1 -5 sqs = 4 1 25 sum_sqs = 30 LAB = 5.4772 % Bruce Mayer * ENGR36 % 25Aug09 * Lec03 % file = VectorLength_0908.m % % Find the length of a Vector AB with %% tail CoOrds = (0, 0, 0) %% tip CoOrds = (2, 1, -5) % % Do Pythagorus in steps vAB = [2 1 -5] % define vector sqs = vAB.*vAB % sq each CoOrd sum_sqs = sum(sqs) % Add all sqs LAB = sqrt(sum_sqs) % Take SQRT of the sum-of-sqs Docsity.com Matrices/Arrays • A MATRIX has multiple ROWS and COLUMNS. For example, the matrix M:             = 15123 948 7316 1042 M  VECTORS are SPECIAL CASES of matrices having ONE ROW or ONE COLUMN.  M contains • FOUR rows • THREE columns Docsity.com Making Matrices • For a small matrix you can type it row by row, separating the elements in a given row with spaces or commas and separating the rows with semicolons. For example, typing       = 7316 1042 A  spaces or commas separate elements in different columns, whereas semicolons separate elements in different rows.  The Command • >>A = [2,4,10;16,3,7];  Generates Matrix Docsity.com M ake M atrix Exam ple >> r1 = [1:5] r1 = 1 2 3 4 5 >> r2 = [6:10] r2 = 6 7 8 9 10 >> r3 = [11:15] r3 = 11 12 13 14 15 >> r4 = [16:20] r4 = 16 17 18 19 20 >> r5 = [21:25] r5 = 21 22 23 24 25 >> A = [r1;r2;r3;r4;r5] A = 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 Docsity.com Array Addressing • The COLON operator SELECTS individual elements, rows, columns, or ''subarrays'' of arrays. Some examples: – v(:) represents all the row or column elements of the vector v – v(2:5) represents the 2nd thru 5th elements; that is v(2), v(3), v(4), v(5). – A(:,3) denotes all the row-elements in the third column of the matrix A. – A(:,2:5) denotes all the row-elements in the 2nd thru 5th columns of A. – A(2:3,1:3) denotes all elements in the 2nd & 3rd rows that are also in the 1st thru 3rd columns. Docsity.com Array Extraction • You can use array indices to extract a smaller array from another array. For example, if you first create the array B             = 1715123 25948 187316 131042 B  to Produce SubMatrix C type C = B(2:3,1:3) Rows 2&3 Cols 1-3       = 948 7316 C Docsity.com Matrix Commands cont Command Description max(A) * Returns the algebraically largest element in A if A is a VECTOR. * Returns a row vector containing the largest elements in each Column if A is a MATRIX. * If any of the elements are COMPLEX, max(A) returns the elements that have the largest magnitudes. [x,k] = max(A) Similar to max(A) but stores the maximum values in the row vector x and their indices in the row vector k. Docsity.com Matrix Commands cont Command Description min(A) and [x,k] = min(A) Like max but returns minimum values. sort(A) Sorts each column of the array A in ascending order and returns an array the same size as A. sum(A) Sums the elements in each column of the array A and returns a row vector containing the sums. Docsity.com Examples: Matrix Commands • Given Matrix A           −= 73 510 26 A >> A = [[6,2]; [-10,5]; [3,7]]; >> size(A) ans = 3 2 >> length(A) ans = 3 >> max(A) ans = 6 7 >> min(A) ans = -10 2 Docsity.com MultiDimensional (≥3) Arrays • 3D (or more-D) Arrays Consist of two-dimensional arrays that ar “layered” to produce a third dimension. – Each “layer” is called a page. 3D pg1 pg2 pg3 pg4 Command Description cat(n,A,B,C, ...) Creates a new array by concatenating the arrays A,B,C, and so on along the dimension n. Docsity.com Vectors Digression • VECTOR ≡ Parameter Possessing Magnitude And Direction, Which Add According To The Parallelogram Law – Examples: Displacements, Velocities, Accelerations • SCALAR ≡ Parameter Possessing Magnitude But Not Direction – Examples: Mass, Volume, Temperature • Vector Classifications – FIXED or BOUND Vectors Have Well Defined Points Of Application That CanNOT Be Changed Without Affecting An Analysis Docsity.com Vectors cont. – FREE Vectors May Be Moved In Space Without Changing Their Effect On An Analysis – SLIDING Vectors May Be Applied Anywhere Along Their Line Of Action Without Affecting the Analysis – EQUAL Vectors Have The Same Magnitude and Direction – NEGATIVE Vector Of a Given Vector Has The Same Magnitude but The Opposite Direction Equal Vectors Negative Vectors Docsity.com Vector Notation • In Print and Handwriting We Must Distinguish Between – VECTORS – SCALARS • These are Equivalent Vector Notations PPP ≡≡≡P • Boldface Preferred for Math Processors • Over Arrow/Bar Used for Handwriting • Underline Preferred for Word Processor Docsity.com Dot Product of 2 Vectors • The SCALAR PRODUCT or DOT PRODUCT Between Two Vectors P and Q Is Defined As ( )resultscalarcosθPQQP =•  PQQP  •=• ( ) 2121 QPQPQQP  •+•=+• ( ) undefined =•• SQP   Scalar-Product Math Properties • ARE Commutative • ARE Distributive • Are NOT Associative – Undefined as (P•S) is NO LONGER a Vector Docsity.com Scalar Product – Cartesian Comps • Scalar Products With Cartesian Components 000 111 =•=•=• =•=•=• ikkjji kkjjii   ( ) ( )kQjQiQkPjPiPQP zyxzyx  ++•++=• 2222 PPPPPP QPQPQPQP zyx zzyyxx =++=• ++=•    MATLAB Makes this Easy with the dot(p,q) Command >> p = [13,-17,-7]; q = [-16,5,23]; >> pq = dot(p,q) pq = -454 >> qp = dot(q,p) qp = -454 >> r = [1 3 5 7]; t = [8,6,4,2]; >> rt = dot(r,t) rt = 60 Docsity.com Vector Prod: Rectangular Comps • Vector Products Of Cartesian Unit Vectors 0 0 0 =×=×−=× −=×=×=× =×−=×=× kkikjjki ijkjjkji jikkijii    ( ) ( )kQjQiQkPjPiPQPV zyxzyx  ++×++=×= ( ) ( ) ( )kQPQPjQPQPiQPQP xyyxzxxzyzzy  −+−+−= zyx zyx QQQ PPP kji QP  =×  Vector Products In Terms Of Rectangular Coordinates • Summarize Using Determinate Notation Docsity.com cross command • 3D Vector (Cross) product Evaluation is Painful – c.f. last Slide • MATLAB makes it easy with the cross(p,q) command – The two Vectors MUST be 3 Dimensional >> p = [13,-17,-7]; >> q = [-16,5,23]; >> pxq = cross(p,q) pxq = -356 -187 -207 >> qxp = cross(q,p) qxp = 356 187 207 Docsity.com Array Addition and Subtraction • Add/Subtract Element-by-Element >> A = [6,-2;10,3]; >> B = [9,8;-12,14]; >> A+B ans = 15 6 -2 17 [ ] [ ] [ ]       − =      − +      − =+ −−=−−−−=− 172 615 1412 89 310 26 16528194113917 BA qp >> p = [17 -9 3]; >> q = [-11,-4,19]; >> p-q ans = 28 -5 -16  Using MATLAB Docsity.com CAVEAT • 2D arrays are described as ROWS x COLUMNS – Elemement of D(2,5) • 2 → ROW-2 • 5 → COL-5 • ROW Vector → COMMAS • COL Vector → SemiColons >> r = [4,73,17] r = 4 73 17 >> c = [13;37;99] c = 13 37 99 Docsity.com Caveat cont • Array operations often CONCATENATE or APPEND – This Can Occur unexpectedly – Use the clear command to start from scratch concatenate con·cat·e·nate ( P ) Pronunciation Key (kn-ktn-t, kn-) tr.v. con·cat·e·nat·ed, con·cat·e·nat·ing, con·cat·e·nates 1. To connect or link in a series or chain. 2. Computer Science. To arrange (strings of characters) into a chained list. adj. (-nt, -nt) 1. Connected or linked in a series. Docsity.com Rectangular Force Components • Using Rt-Angle Parallelogram Resolve F into Perpendicular Components yx FFF += yxyx FF jiFFF +=+=  Define Perpendicular UNIT Vectors Which Are Parallel To The Axes  Vectors May then Be Expressed As Products Of The Unit Vectors With The SCALAR MAGNITUDES Of The Vector Components Docsity.com 3D Rectangular Components • Resolve Fh into rectangular components  Resolve F into horizontal and vertical components. yh yy FF FF F θ θ sin cos = = = F φθ φ φθ φ sinsin sin cossin cos y hz y hx F FF F FF = = = =  The vector F is contained in the plane OBAC. Docsity.com 3D Rectangular Comp. cont. ( ) kjiλ λ kji kjiF zyx zyx zyx zzyyxx F F FFF FFFFFF θθθ θθθ θθθ coscoscos coscoscos coscoscos ++= = ++= ++= ===  With the angles between F and the axes  λ is a UNIT VECTOR along the line of action of F, and cosθx, cosθy, and cosθz, are the DIRECTION COSINES for F Docsity.com 2-Pt Direction Cosines • Consider Force, F, Directed Between Pts – M(x1,y1,z1) – N(x2,y2,z2) • The Vector MN Joins These Points with Scalar Components dx, dy, dz ( ) d d d d d d d FdF d Fd F d FdF ddd d F dddd z z y y x x z z y y x x zyx zyx === === ++== ++== θθθ coscoscos 1 222 kjiλF d 12 1212 zzd yydxxd dddMN z yx zyx −= −=−= ++== kjid Docsity.com