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Algorithm for matrix inversion

  1. Sep 17, 2003 #1
    Anybody know of a link to a page that describes an algorithm for matrix inversion. My old linear algebra book describes a 'by hand' method, but it's unsuitable for automating.
     
  2. jcsd
  3. Sep 17, 2003 #2

    Hurkyl

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    Gaussian elimination isn't that bad, is it? :smile:


    Important concerns include the size of your matrix, any qualitative properties it may have (such as sparse, symmetric, or banded), and what you want to do with the inverse.

    For example, the conjugate gradient method is well suited for solving the equation Ax=b when A is large sparse matrix, but it won't explicitly compute A inverse.
     
    Last edited: Sep 17, 2003
  4. Sep 17, 2003 #3

    chroot

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    Yeah, Gaussian elimination is the only method that's gauranteed to work on all matrices, but it is the slowest.

    - Warren
     
  5. Sep 17, 2003 #4

    Integral

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    Rather then do a web search, I searched my book shelf. These routines are in Fortran (thats how old my book shelf is!) Not sure if this is the best format, but you will have to translate to a modern language anyway. I believe that the syntax is simple enough that you should (with fundamental programing knowledge) be able to figure out what is happening. I think I have edited out all of the OCR glitches, my software wanted to read an = as a z, so if there is a wierd z that makes no sense floating around replace it with an =.

    These were lifted from Elementary Numerical Analysis by Conte and de Boor.
    I will do my best to answer any questions.

    BTW
    In the text preceding the inversion routine the authors say

    SUBROUTINE SUBST ( W, IPIVOT, B, N, X )
    INTEGER IPIVOT(N) , I,IP,J
    REAL B(N) ,W(N,N) ,X(N), SUM
    C****** I N PUT ******
    C W, IPIVOT, N ARE AS ON OUTPUT FROM F ACT 0 R , APPLIED TO THE
    C MATRIX A OF ORDER N.
    C B IS AN N-VECTOR, GIVING THE RIGHT SIDE OF THE SYSTEM TO BE SOLVED.
    C****** 0 U T PUT ******
    C X IS THE N-VECTOR SATISFYING A*X . B.
    C****** MET HOD ******
    C ALGORITHM 4.4 IS USED, I.E., THE FACTORIZATION OF A CONTAINED IN
    C W AND IPIVOT (AS GENERATED IN FACTOR) IS USED TO SOLVE A*X = B
    C FOR X BY SOLVING TWO TRIANGULAR SYSTEMS.
    C
    IF (N .LE. 1) THEN
    X(1) = B(1)/W(1,1)
    RETURN
    END IF
    IP = IPIVOT(1)
    X(1) = B(IP)
    DO 15 I=2,N
    SUM = 0.
    DO 14 J=I,I-l
    14 SUM = W(I,J)*X(J) + SUM
    IP z IPIVOT(I)
    15 X(I) = B(IP) - SUM
    C
    X(N) = X(N)/W(N,N)
    DO 20 I=N-l,I,-1
    SUM = 9.
    DO 19 J=I+l,N
    19 SUM = W(I,J)*X(J) + SUM
    20 X(I) = (X(I) - SUM)/W(I,I)
    RETURN
    END

    SUBROUTINE FACTOR ( W, N, D, IPIVOT, IFLAG )
    INTEGER IFLAG,IPIVOT(N) , I,ISTAR,J,K
    REAL D(N) ,W(N,N), AWIKOD,COLMAX,RATIO,ROWMAX,TEMP
    **** INPUT ******
    C W ARRAY OF SIZE (N,N) CONTAINING THE MATRIX A OF ORDER N TO BE
    C FACTORED.
    C N THE ORDER OF THE MATRIX
    C***** WORK AREA ******
    C D A REAL VECTOR OF LENGTH N, TO HOLD ROW SIZES
    C***** 0UTPUT ******
    C W ARRAY OF SIZE (N,N) CONTAINING THE LU FACTORIZATION OF P*A FOR
    C SOME PERMUTATION MATRIX P SPECIFIED BY IPIVOT.
    C IPIVOT INTEGER VECTOR OF LENGTH N INDICATING THAT ROW IPIVOT(K)
    C WAS USED TO ELIMINATE X(K) , K-l,...,N .
    C IFLAG AN INTEGER,
    C = 1, IF AN EVEN NUMBER OF INTERCHANGES WAS CARRIED OUT,
    C = -1,IF AN ODD NUMBER OF INTERCHANGES WAS CARRIED OUT,
    C = 0, IF THE UPPER TRIANGULAR FACTOR HAS ONE OR MORE ZERO DIA-
    C GONAL ENTRIES.
    C THUS, DETERMINANT (A) - IFLAG*W(l,l)*...*W(N,N) .
    C IF IFLAG .NE. 0, THEN THE LINEAR SYSTEM A*X 8 B CAN BE SOLVED FOR
    C X BY A
    C CALL SUBST (W, IPIVOT, B, N, X )
    C**** METHOD ******
    C THE PROGRAM FOLLOWS ALGORITHM 4.2, USING SCALED PARTIAL PIVOTING.
    C
    IFLAG = 1
    C INITIALIZE IPIVOT, D
    DO 9 I-l,N
    IPIVOT(I) = I
    ROWMAX = 0.
    DO 5 J=l,N
    5 ROWMAX = AMAXl(ROWMAX,ABS(W(I,J)))
    IF (ROWMAX .Eq. 0.) THEN
    IFLAG = 0
    ROWMAX = 1.
    END IF
    9 D(I) = ROWMAX
    IF (N .LE. 1) RETURN
    C FACTORIZATION
    DO 20 K=l,N-l
    C DETERMINE PIVOT ROW, THE ROW ISTAR.
    COLMAX = ABS(W(K,K)/D(K)
    ISTAR = K
    DO 13 I=K+l,N
    AWIKOD = ABS(W(I,K))/D(I)
    IF (AWIKOD .GT. COLMAX) THEN
    COLMAX = AWIKOD
    ISTAR = I
    END IF
    13 CONTINUE
    IF (COLMAX .EO. 0.) THEN
    IFLAG = 0
    ELSE
    IF (ISTAR .GT. K) THEN
    C MAKE K THE PIVOT ROW BY INTERCHANGING IT WITH
    C THE CHOSEN ROW ISTAR.
    IFLAG = -IFLAG
    I = IPIVOT(ISTAR)
    IPIVOT(ISTAR) = IPIVOT(K)
    IPIVOT(K) = I
    TEMP = D(ISTAR)
    D(ISTAR) = D(K)
    D(K) = TEMP
    DO 15 J=l,N
    TEMP = W(ISTAR,J)
    W(ISTAR,J) = W(K,J)
    15 W(K,J) = TEMP
    END IF
    C ELIMINATE X(K) FROM ROWS K+l,...,N.
    l6 DO 19 I=K+l,N
    W(I,K) = W(I,K)/W(K,K)
    RATIO = W(I,K)
    DO 19 J=K=1,N
    W(I,J) = W(I,J) - RATIO*W(K,J)
    19 CONTINUE
    END IF
    20 CONTINUE
    IF (W(N,N) .EQ. 0.) IFLAG -0
    RETURN
    END

    C PROGRAM FOR CALCULATING THE INVERSE OF A GIVEN MATRIX
    C CALLS FACT0R, SUBST.
    PARAMETER NMAX=30,NMAXSQ=NMAX*NMAX
    INTEGER I,IBEG,IFLAG,IPIVOT(NMAX) ,J,N,NSQ
    REAL A(NMAXSQ) ,AINV(NMAXSQ) ,B(NMAX)
    1 READ 501, N
    501 FORMAT(I2)
    IF (N .LT. 1 .OR. N .GT. NMAX) STOP
    C READ IN MATRIX ROW BY ROW
    NSQ = N*N
    DO 10 I=I,N
    18 READ 510, (A(J) ,J=I,NSQ,N)
    510 FORMAT(5EI5.7)
    C
    CALL FACTOR ( A, N, B, IPIVOT, IFLAG )
    IF (IFLAG .EQ. 0) THEN
    PRINT 611
    611 FORMAT('IMATRIX IS SINGULAR')
    GO TO 1
    END IF
    DO 21 I=l,N
    21 B(I) = 0.
    IBEG = 1
    DO 30 J=I,N
    B(J) = 1.
    CALL SUBST ( A, IPIVOT, B, N, AINV(IBEG) )
    B(J) = 0.
    38 IBEG = IBEG + N
    PRINT 630
    630 FORMAT('ITHE COMPUTED INVERSE IS '//)
    DO 31 I=l,N
    31 PRINT 631, I, (AINV(J) ,J-I,NSQ,N)
    631 FORMAT('0ROW ',I2,8EI5.7/(7X,8EI5.7))
    GO TO 1
    END
     
    Last edited: Sep 17, 2003
  6. Sep 17, 2003 #5

    chroot

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    Well, Fortran is pretty lame overall.

    Numerical Recipes also has a lot of good material on the topic, in both fortran and C++. And, as quite a nice gift to the scientific community, the books are available in their entirety online:

    http://ww.nr.com

    - Warren
     
  7. Sep 17, 2003 #6

    Integral

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    Warren,
    The routines deserve a look inspite of the language, the above text generally presented very nice efficient routines, even if the language is archaic. :)

    BTW: you may want to edit your link www usually works better then ww!
     
  8. Sep 17, 2003 #7
    Thanks to all of you for the info. Do worry about it being in FORTRAN, unfortunately I'm old enough to be able to read it. :smile:
    I was after the algorithm, I can always code it up into Java.

    I throw and fire pottery for a hobby, and I've always had it in the back of my mind to write a glaze manipulation program. All the important properties of the ingredients are linearly translated into the glaze, so by using AX = B where A is a vector of ingredient percentages, X is the component/properties matrix, and B is the glaze component/properties vector, I can make changes in glaze properties (B), derive X-1, then come up with an new set of proportions corrosponding to the glaze I want.

    A mundane application, but it's a real pain when the thermal coefficient of expansion for the glaze and clay body don't match.
     
  9. Sep 18, 2003 #8

    great site....
    I'we wanted a good book on this ever since I studied this course in college (the profeesor was such a looser though I couldn't understand much from him).
    My eternal gratitude....:smile:
    PS: edited the above link....
     
  10. Sep 17, 2004 #9

    mathwonk

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    for matrices near the identity matrix, i.e. of form I-f for small f, what about the geometric series?

    aha! i got the center of the earth, or the beginning of time. the goal of all paleontologists!
     
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