ECE210a_lecture9_small

ECE210a_lecture9_small - Numerical aspects t-Digit...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Gram-Schmidt Gram-Schmidt example Consider finding an orthonormal basis from the vectors, x 1 = 1 1 × 10 3 1 × 10 3 , x 2 = 1 1 × 10 3 0 , x 3 = 1 0 1 × 10 3 , using 3-digit arithmetic. bardbl x 1 bardbl = radicalbig 1 + 1 × 10 6 + 1 × 10 6 = 1 , so q 1 = x 1 ; Now, q T 1 x 2 = bracketleftbig 1 1 × 10 3 1 × 10 3 bracketrightbig 1 1 × 10 3 0 = 1 . To find the direction of the second orthonormal vector, ˆ q 2 = x 2 −( q 1 ,x 2 ) q 1 = 1 1 × 10 3 0 1 × 1 1 × 10 3 1 × 10 3 = 0 0 1 × 10 3 . Roy Smith: ECE 210a: 9 .2 Numerical aspects t -Digit Arithmetic. Examining numerical aspects. Given x ∈R , define the t -digit floating point, representation, f t ( x ) = 0 .d 1 d 2 ... d t × 10 e , where the digits, d i , and the exponent, e , minimize | x f t ( x ) | . If f t ( x ) is not unique, round away from zero. Unfortunate properties 1. f t ( x + y ) negationslash = f t ( x ) + f t ( y ). 2. f t ( xy ) negationslash = f t ( x ) f t ( y ). Consider 2-digit arithmetic with x = 21 / 2 and y = 11 / 2. Roy Smith: ECE 210a: 9 .1
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Gram-Schmidt Floating arithmetic example: Three digit arithmetic gives: q 1 = 1 1 × 10 3 1 × 10 3 , q 2 = 0 0 1 , q 3 = 0 0 . 709 0 . 709 bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright . These two vectors are at 45 degrees! Definitely NOT orthogonal. The problem arises from the fact that the first component in each vector dominates the differences in the second two components in the calculation. Modified Gram-Schmidt In the Gram-Schmidt procedure we have, q k = ( I Q k 1 Q k 1 ) x k vextenddouble vextenddouble ( I Q k 1 Q k 1 ) x k vextenddouble vextenddouble , Q k 1 = bracketleftbig q 1 ... q k 1 bracketrightbig , Q 0 = 0 . Consider expressing the calculation of q k as a product of matrix operations. Define the projections, P 0 = I, P i = I q i q i ←− removes the part aligned with q i Roy Smith: ECE 210a: 9 .4 Gram-Schmidt Normalizing ˆ q 2 gives, q 2 = 0 0 1 . To get the last vector, ˆ q 3 = x 3 −( q 1 ,x 3 ) q 1 −( q 2 ,x 3 ) q 2 , and, ( q 1 ,x 3 ) = q T 1 x 3 = bracketleftbig 1 1 × 10 3 1 × 10 3 bracketrightbig 1 0 1 × 10 3 = 1 ( q 2 ,x 3 ) = q T 2 x 3 = bracketleftbig 0 0 1 bracketrightbig 1 0 1 × 10 3 = 1 × 10 3 .
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern