ekiim's blog

Exercises

  1. Apply elimination and back-substitution to solve 2u3v=32u5v+w=72uv3w=5 2u -3v = 3 2u -5v + w = 7 2u - v -3w = 5 {.aligned} Where are the pivots? List the three operations in which a multiple of one row is substracted from another.

  2. For the system u+v+w=2y+3v+3w=0y+3v+5w=2, u + v + w = 2 y + 3v + 3w = 0 y + 3v + 5w = 2, {.aligned} what is the triangular system after forward elimination, and what is the solution?

  3. Solve the system and find the pivots when 2uv=0u+2vw=0v+2wz=0w+2z=5 2u - v = 0 -u + 2v - w = 0 - v +2w - z = 0 - w +2z = 5 {.aligned} You may carry the right side as fifth column (and omit wrigint u,v,w,zu, v, w, z until the solution at the end).

  4. Apply Elimination to the systemu+v+w=23u+3vw=6yv+w=1 u + v + w = -2 3u + 3v - w = 6 y - v + w = -1 {.aligned} When a zaero arises in the pivot position, exchange that equation for the one below it and proceed. What coefficient of vv in the third equation, in place of the present 1-1, would make it impossible to procceed — and force elimination to breakd down?

  5. Solve by elimination the system of two equations xy=03x+6y=18 x - y = 0 3x + 6y = 18 {.aligned} Draw a graph representing each equation as straight lines in the xyx-y plane; the lines intersect at the solution. Also add one more line – the graph of the new equation which arises after elimination.

  6. Find three values of aa for which elimination breaks down, temporarly or permanently, in au+v=14u+av=2 au + v = 1 4u + av = 2 {.aligned} Breakdown at the first step can be fixed by exchanging rows – but not breakdown at last step.

    1. If the first two rows of AA are the same, when will elimination discover that AA is singular? Do a 3 by 3 example, allowing row exchanges.
    2. If the first two columns of AA are the same, when will elimination discover that AA is singular? Do a 3 by 3 example, allowing column exchanges.
  7. How many multiplications-substractions would it take to solve a system of order n=600n = 600?. How Many seconds, on a PC that can do 80008000 a seconds or a VAX that can do 80,00080,000, or a CRAY X-MP/2 that can do 12 million? (Thoes are in double precision — I think the CRAY is cheapest if you can afford it.)

  8. True or false:

    1. If the third equation starts with a zero coefficient (it begins with 0u0u) then no multiple of equation 1 will be substracted from equation 3.
    2. If the third equation has a zero in the second coefficient (it contains (0v) then no multiple of equation 2 will be substracted from equation 3.
    3. If the third equation contains 0u0u and 0v0v, then no multiple of equation 1, or equation 2 will be substracted from equestion 3.
  9. (very optional) Normally the multiplitaction of two complex numbers (a+ib)(c+id)=(acbd)+i(bc+ad) (a + ib)(c + id) = (ac - bd) + i(bc + ad) involves the four seperated multiplications ac,bd,bc,adac, bd, bc, ad. Ingoring i, can you compute the quantities acbdac-bd and bc+adbc+ad with only three multiplications? (You may do additions, such as forming a,ba, b, before multiplying without any penalty. We note hoever that additions take six cloc cycles on a CRAY X-MP/48 and multiplications is only one more time.)

  10. Use elimination to solve

    u+v+w=b1v+w=b2w=b3\begin{aligned} u + v + w &= b_1 \\ v + w &= b_2 \\ w &= b_3 \end{aligned}

    And

    u+v+w=7u+2v+2w=102u+2v4w=3\begin{aligned} u + v + w &= 7 \\ u + 2v + 2w &= 10 \\ 2u + 2v - 4w &= 3 \\ \end{aligned}

The final exercises gives experience in setting up linear equations. suppose that (a) of those start a year in California, 80 percent stay in and 20 percent move out; (b) of those start a year in California, 90 percent stay in and 10 percent move out; If we know the situation at the beginning, say 200 million outside and 30 million in, the it is easy to find the number uu and vv that are outisde and inside at the end: .9(200,000,000)+.2(30,000,000)=u.1(200,000,000)+.8(30,000,000)=v\begin{aligned} .9(200,000,000) + .2(30,000,000) &= u\\ .1(200,000,000) + .8(30,000,000) &= v \end{aligned} The real problem is to go backwards, and compute the start from the finish.

  1. If u=u = 200 million and v=v = 30 million at the end, set up (without solving) the equations to find uu, and vv, at the beginning.
  2. If uu and vv at the end are teh same as uu and vv at the beginning what equations do you get? What is the ratio of uu to vv in this “steady state”?