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Text File  |  1998-10-30  |  18.8 KB  |  265 lines
  1.  
  2.  
  3.  
  4. ____TTTTRRRRMMMMVVVV,,,,____TTTTRRRRSSSSVVVV((((3333FFFF))))                                                ____TTTTRRRRMMMMVVVV,,,,____TTTTRRRRSSSSVVVV((((3333FFFF))))
  5.  
  6.  
  7.  
  8. NNNNAAAAMMMMEEEE
  9.      dtrmv, strmv, ztrmv, ctrmv, dtrsv, strsv, ztrsv, ctrsv - BLAS Level Two
  10.      Matrix-Vector Product and Solution of system of equations.
  11.  
  12.  
  13. FFFFOOOORRRRTTTTRRRRAAAANNNN 77777777 SSSSYYYYNNNNOOOOPPPPSSSSIIIISSSS
  14.      ssssuuuubbbbrrrroooouuuuttttiiiinnnneeee ddddttttrrrrmmmmvvvv(((( uuuupppplllloooo,,,, ttttrrrraaaannnnssss,,,, ddddiiiiaaaagggg,,,, nnnn,,,, aaaa,,,, llllddddaaaa,,,, xxxx,,,, iiiinnnnccccxxxx ))))
  15.      ssssuuuubbbbrrrroooouuuuttttiiiinnnneeee ddddttttrrrrssssvvvv(((( uuuupppplllloooo,,,, ttttrrrraaaannnnssss,,,, ddddiiiiaaaagggg,,,, nnnn,,,, aaaa,,,, llllddddaaaa,,,, xxxx,,,, iiiinnnnccccxxxx ))))
  16.            cccchhhhaaaarrrraaaacccctttteeeerrrr****1111        uuuupppplllloooo,,,, ttttrrrraaaannnnssss,,,, ddddiiiiaaaagggg
  17.            iiiinnnntttteeeeggggeeeerrrr            nnnn,,,, llllddddaaaa,,,, iiiinnnnccccxxxx
  18.            ddddoooouuuubbbblllleeee pppprrrreeeecccciiiissssiiiioooonnnn   aaaa(((( llllddddaaaa,,,,****)))),,,, xxxx((((****))))
  19.  
  20.      ssssuuuubbbbrrrroooouuuuttttiiiinnnneeee ssssttttrrrrmmmmvvvv(((( uuuupppplllloooo,,,, ttttrrrraaaannnnssss,,,, ddddiiiiaaaagggg,,,, nnnn,,,, aaaa,,,, llllddddaaaa,,,, xxxx,,,, iiiinnnnccccxxxx ))))
  21.      ssssuuuubbbbrrrroooouuuuttttiiiinnnneeee ssssttttrrrrssssvvvv(((( uuuupppplllloooo,,,, ttttrrrraaaannnnssss,,,, ddddiiiiaaaagggg,,,, nnnn,,,, aaaa,,,, llllddddaaaa,,,, xxxx,,,, iiiinnnnccccxxxx ))))
  22.            cccchhhhaaaarrrraaaacccctttteeeerrrr****1111        uuuupppplllloooo,,,, ttttrrrraaaannnnssss,,,, ddddiiiiaaaagggg
  23.            iiiinnnntttteeeeggggeeeerrrr            nnnn,,,, llllddddaaaa,,,, iiiinnnnccccxxxx
  24.            rrrreeeeaaaallll               aaaa(((( llllddddaaaa,,,,****)))),,,, xxxx((((****))))
  25.  
  26.      ssssuuuubbbbrrrroooouuuuttttiiiinnnneeee zzzzttttrrrrmmmmvvvv(((( uuuupppplllloooo,,,, ttttrrrraaaannnnssss,,,, ddddiiiiaaaagggg,,,, nnnn,,,, aaaa,,,, llllddddaaaa,,,, xxxx,,,, iiiinnnnccccxxxx ))))
  27.      ssssuuuubbbbrrrroooouuuuttttiiiinnnneeee zzzzttttrrrrssssvvvv(((( uuuupppplllloooo,,,, ttttrrrraaaannnnssss,,,, ddddiiiiaaaagggg,,,, nnnn,,,, aaaa,,,, llllddddaaaa,,,, xxxx,,,, iiiinnnnccccxxxx ))))
  28.            cccchhhhaaaarrrraaaacccctttteeeerrrr****1111        uuuupppplllloooo,,,, ttttrrrraaaannnnssss,,,, ddddiiiiaaaagggg
  29.            iiiinnnntttteeeeggggeeeerrrr            nnnn,,,, llllddddaaaa,,,, iiiinnnnccccxxxx
  30.            ddddoooouuuubbbblllleeee ccccoooommmmpppplllleeeexxxx     aaaa(((( llllddddaaaa,,,,****)))),,,, xxxx((((****))))
  31.  
  32.      ssssuuuubbbbrrrroooouuuuttttiiiinnnneeee ccccttttrrrrmmmmvvvv(((( uuuupppplllloooo,,,, ttttrrrraaaannnnssss,,,, ddddiiiiaaaagggg,,,, nnnn,,,, aaaa,,,, llllddddaaaa,,,, xxxx,,,, iiiinnnnccccxxxx ))))
  33.      ssssuuuubbbbrrrroooouuuuttttiiiinnnneeee ccccttttrrrrssssvvvv(((( uuuupppplllloooo,,,, ttttrrrraaaannnnssss,,,, ddddiiiiaaaagggg,,,, nnnn,,,, aaaa,,,, llllddddaaaa,,,, xxxx,,,, iiiinnnnccccxxxx ))))
  34.            cccchhhhaaaarrrraaaacccctttteeeerrrr****1111        uuuupppplllloooo,,,, ttttrrrraaaannnnssss,,,, ddddiiiiaaaagggg
  35.            iiiinnnntttteeeeggggeeeerrrr            nnnn,,,, llllddddaaaa,,,, iiiinnnnccccxxxx
  36.            ccccoooommmmpppplllleeeexxxx            aaaa(((( llllddddaaaa,,,,****)))),,,, xxxx((((****))))
  37.  
  38.  
  39.  
  40. CCCC SSSSYYYYNNNNOOOOPPPPSSSSIIIISSSS
  41.      vvvvooooiiiidddd ddddttttrrrrmmmmvvvv(((( uuuupppplllloooo,,,, ttttrrrraaaannnnssss,,,, ddddiiiiaaaagggg,,,, nnnn,,,, aaaa,,,, llllddddaaaa,,,, xxxx,,,, iiiinnnnccccxxxx ))))
  42.      vvvvooooiiiidddd ddddttttrrrrssssvvvv(((( uuuupppplllloooo,,,, ttttrrrraaaannnnssss,,,, ddddiiiiaaaagggg,,,, nnnn,,,, aaaa,,,, llllddddaaaa,,,, xxxx,,,, iiiinnnnccccxxxx ))))
  43.            MMMMaaaattttrrrriiiixxxxTTTTrrrriiiiaaaannnngggglllleeee        uuuupppplllloooo;;;;
  44.            MMMMaaaattttrrrriiiixxxxTTTTrrrraaaannnnssssppppoooosssseeee       ttttrrrraaaannnnssss;;;;
  45.            MMMMaaaattttrrrriiiixxxxUUUUnnnniiiittttTTTTrrrriiiiaaaannnngggguuuullllaaaarrrr  ddddiiiiaaaagggg;;;;
  46.            IIIInnnntttteeeeggggeeeerrrr               nnnn,,,, llllddddaaaa,,,, iiiinnnnccccxxxx;;;;
  47.            ddddoooouuuubbbblllleeee                ((((****aaaa))))[[[[llllddddaaaa****nnnn]]]],,,, ((((****xxxx))))[[[[ nnnn ]]]];;;;
  48.  
  49.      vvvvooooiiiidddd ssssttttrrrrmmmmvvvv(((( uuuupppplllloooo,,,, ttttrrrraaaannnnssss,,,, ddddiiiiaaaagggg,,,, nnnn,,,, aaaa,,,, llllddddaaaa,,,, xxxx,,,, iiiinnnnccccxxxx ))))
  50.      vvvvooooiiiidddd ssssttttrrrrssssvvvv(((( uuuupppplllloooo,,,, ttttrrrraaaannnnssss,,,, ddddiiiiaaaagggg,,,, nnnn,,,, aaaa,,,, llllddddaaaa,,,, xxxx,,,, iiiinnnnccccxxxx ))))
  51.            MMMMaaaattttrrrriiiixxxxTTTTrrrriiiiaaaannnngggglllleeee        uuuupppplllloooo;;;;
  52.            MMMMaaaattttrrrriiiixxxxTTTTrrrraaaannnnssssppppoooosssseeee       ttttrrrraaaannnnssss;;;;
  53.            MMMMaaaattttrrrriiiixxxxUUUUnnnniiiittttTTTTrrrriiiiaaaannnngggguuuullllaaaarrrr  ddddiiiiaaaagggg;;;;
  54.            IIIInnnntttteeeeggggeeeerrrr               nnnn,,,, llllddddaaaa,,,, iiiinnnnccccxxxx;;;;
  55.            ffffllllooooaaaatttt                 ((((****aaaa))))[[[[llllddddaaaa****nnnn]]]],,,, ((((****xxxx))))[[[[ nnnn ]]]];;;;
  56.  
  57.      vvvvooooiiiidddd zzzzttttrrrrmmmmvvvv(((( uuuupppplllloooo,,,, ttttrrrraaaannnnssss,,,, ddddiiiiaaaagggg,,,, nnnn,,,, aaaa,,,, llllddddaaaa,,,, xxxx,,,, iiiinnnnccccxxxx ))))
  58.      vvvvooooiiiidddd zzzzttttrrrrssssvvvv(((( uuuupppplllloooo,,,, ttttrrrraaaannnnssss,,,, ddddiiiiaaaagggg,,,, nnnn,,,, aaaa,,,, llllddddaaaa,,,, xxxx,,,, iiiinnnnccccxxxx ))))
  59.            MMMMaaaattttrrrriiiixxxxTTTTrrrriiiiaaaannnngggglllleeee        uuuupppplllloooo;;;;
  60.  
  61.  
  62.  
  63.                                                                         PPPPaaaaggggeeee 1111
  64.  
  65.  
  66.  
  67.  
  68.  
  69.  
  70. ____TTTTRRRRMMMMVVVV,,,,____TTTTRRRRSSSSVVVV((((3333FFFF))))                                                ____TTTTRRRRMMMMVVVV,,,,____TTTTRRRRSSSSVVVV((((3333FFFF))))
  71.  
  72.  
  73.  
  74.            MMMMaaaattttrrrriiiixxxxTTTTrrrraaaannnnssssppppoooosssseeee       ttttrrrraaaannnnssss;;;;
  75.            MMMMaaaattttrrrriiiixxxxUUUUnnnniiiittttTTTTrrrriiiiaaaannnngggguuuullllaaaarrrr  ddddiiiiaaaagggg;;;;
  76.            IIIInnnntttteeeeggggeeeerrrr               nnnn,,,, llllddddaaaa,,,, iiiinnnnccccxxxx;;;;
  77.            ZZZZoooommmmpppplllleeeexxxx               ((((****aaaa))))[[[[llllddddaaaa****nnnn]]]],,,, ((((****xxxx))))[[[[ nnnn ]]]];;;;
  78.  
  79.      vvvvooooiiiidddd ccccttttrrrrmmmmvvvv(((( uuuupppplllloooo,,,, ttttrrrraaaannnnssss,,,, ddddiiiiaaaagggg,,,, nnnn,,,, aaaa,,,, llllddddaaaa,,,, xxxx,,,, iiiinnnnccccxxxx ))))
  80.      vvvvooooiiiidddd ccccttttrrrrssssvvvv(((( uuuupppplllloooo,,,, ttttrrrraaaannnnssss,,,, ddddiiiiaaaagggg,,,, nnnn,,,, aaaa,,,, llllddddaaaa,,,, xxxx,,,, iiiinnnnccccxxxx ))))
  81.            MMMMaaaattttrrrriiiixxxxTTTTrrrriiiiaaaannnngggglllleeee        uuuupppplllloooo;;;;
  82.            MMMMaaaattttrrrriiiixxxxTTTTrrrraaaannnnssssppppoooosssseeee       ttttrrrraaaannnnssss;;;;
  83.            MMMMaaaattttrrrriiiixxxxUUUUnnnniiiittttTTTTrrrriiiiaaaannnngggguuuullllaaaarrrr  ddddiiiiaaaagggg;;;;
  84.            IIIInnnntttteeeeggggeeeerrrr               nnnn,,,, llllddddaaaa,,,, iiiinnnnccccxxxx;;;;
  85.  
  86.            CCCCoooommmmpppplllleeeexxxx               ((((****aaaa))))[[[[llllddddaaaa****nnnn]]]],,,, ((((****xxxx))))[[[[ nnnn ]]]];;;;
  87.  
  88.  
  89.  
  90. DDDDEEEESSSSCCCCRRRRIIIIPPPPTTTTIIIIOOOONNNN
  91.      ddddttttrrrrmmmmvvvv, ssssttttrrrrmmmmvvvv, zzzzttttrrrrmmmmvvvv and ccccttttrrrrmmmmvvvv perform one of the matrix-vector operations
  92.  
  93.            x := A*x,   or   x := A'*x,   or   x := conjg( A' )*x,
  94.  
  95.      where x is an n element vector and  A is an n by n unit, or non-unit,
  96.      upper or lower triangular matrix.
  97.  
  98.      ddddttttrrrrssssvvvv, ssssttttrrrrssssvvvv, zzzzttttrrrrssssvvvv and ccccttttrrrrssssvvvv solve one of the systems of equations
  99.  
  100.            A*x = b,   or   A'*x = b,   or   conjg( A' )*x = b,
  101.  
  102.      where b and x are n element vectors and A is an n by n unit, or non-unit,
  103.      upper or lower triangular matrix.  No test for singularity or near-
  104.      singularity is included in these routines. Such tests must be performed
  105.      before calling these routines.
  106.  
  107.  
  108. PPPPAAAARRRRAAAAMMMMEEEETTTTEEEERRRRSSSS
  109.      uuuupppplllloooo    On entry, uuuupppplllloooo specifies whether the matrix is an upper or lower
  110.              triangular matrix as follows:
  111.  
  112.                   FFFFOOOORRRRTTTTRRRRAAAANNNN
  113.                   uplo = 'U' or 'u'        A is an upper triangular matrix.
  114.                   uplo = 'L' or 'l'        A is a lower triangular matrix.
  115.  
  116.                   CCCC
  117.                   uplo = UpperTriangle     A is an upper triangular matrix.
  118.                   uplo = LowerTriangle     A is a lower triangular matrix.
  119.  
  120.              Unchanged on exit.
  121.  
  122.      ttttrrrraaaannnnssss   On entry, ttttrrrraaaannnnssss ssssppppeeeecccciiiiffffiiiieeeessss tttthhhheeee ooooppppeeeerrrraaaattttiiiioooonnnn ttttoooo bbbbeeee
  123.  
  124.                   FFFFOOOORRRRTTTTRRRRAAAANNNN
  125.                   trans = 'N' or 'n'       x := A*x /  A*x = b.
  126.  
  127.  
  128.  
  129.                                                                         PPPPaaaaggggeeee 2222
  130.  
  131.  
  132.  
  133.  
  134.  
  135.  
  136. ____TTTTRRRRMMMMVVVV,,,,____TTTTRRRRSSSSVVVV((((3333FFFF))))                                                ____TTTTRRRRMMMMVVVV,,,,____TTTTRRRRSSSSVVVV((((3333FFFF))))
  137.  
  138.  
  139.  
  140.                  trans = 'T' or 't'        x := A'*x / A'*x = b.
  141.                   trans = 'C' or 'c'       x := conjg( A' )*x  /
  142.                                                 conjg( A' )*x = b.
  143.  
  144.                   CCCC
  145.                   trans = NoTranspose           x := A*x /  A*x = b.
  146.                   trans = Transpose             x := A'*x / A'*x = b.
  147.                   trans = ConjugateTranspose    x := conjg( A' )*x  /
  148.                                                 conjg( A' )*x = b.
  149.  
  150.              For real value matrices, ttttrrrraaaannnnssss='C' and ttttrrrraaaannnnssss='T' has the same
  151.              meaning.
  152.  
  153.              Unchanged on exit.
  154.  
  155.      ddddiiiiaaaagggg    On entry, ddddiiiiaaaagggg specifies whether or not A is unit triangular as
  156.              follows:
  157.  
  158.                   FFFFOOOORRRRTTTTRRRRAAAANNNN
  159.                   diag = 'U' or 'u'   A is assumed to be unit triangular.
  160.                   diag = 'N' or 'n'   A is not assumed to be unit triangular.
  161.  
  162.                   CCCC
  163.                   diag = UnitTriangular    A is assumed to be unit
  164.                                            triangular.
  165.                   diag = NotUnitTriangular A is not assumed to be unit
  166.                                            triangular.
  167.  
  168.              Unchanged on exit.
  169.  
  170.      nnnn       On entry, nnnn specifies the order of the matrix A. nnnn must be at
  171.              least zero.
  172.              Unchanged on exit.
  173.  
  174.      aaaa       An array containing the matrix A.
  175.  
  176.              FFFFOOOORRRRTTTTRRRRAAAANNNN
  177.              Array of dimension (lda, n).
  178.  
  179.              CCCC
  180.              A pointer to an array of size lda*n.
  181.              See note below about array storage convention for C.
  182.  
  183.              Before entry with uuuupppplllloooo ==== ''''UUUU'''' oooorrrr ''''uuuu'''' oooorrrr , the elements of the
  184.              array aaaa corresponding to the leading n by n upper triangular part
  185.              of the matrix A must contain the upper triangular matrix and the
  186.              elements corresponding to the strictly lower triangular part of A
  187.              are not referenced.
  188.  
  189.              Before entry with uuuupppplllloooo ==== ''''LLLL'''' oooorrrr ''''llll'''' oooorrrr , the elements
  190.              corresponding to the leading n by n lower triangular elements of
  191.              the matrix A  must contain the lower triangular matrix and the
  192.  
  193.  
  194.  
  195.                                                                         PPPPaaaaggggeeee 3333
  196.  
  197.  
  198.  
  199.  
  200.  
  201.  
  202. ____TTTTRRRRMMMMVVVV,,,,____TTTTRRRRSSSSVVVV((((3333FFFF))))                                                ____TTTTRRRRMMMMVVVV,,,,____TTTTRRRRSSSSVVVV((((3333FFFF))))
  203.  
  204.  
  205.  
  206.              corresponding strictly upper triangular part of the matrix A is
  207.              not referenced.
  208.  
  209.              Note that when ddddiiiiaaaagggg ==== ''''UUUU'''' oooorrrr ''''uuuu'''' oooorrrr , the elements of _a
  210.              corresponding to the diagonal elements of the matrix A are not
  211.              referenced either, but are assumed to be unity.
  212.  
  213.              Unchanged on exit.
  214.  
  215.      llllddddaaaa     On entry, llllddddaaaa specifies the first dimension of A as declared in
  216.              the calling (sub) program.  llllddddaaaa must be at least max( 1, n ).
  217.              Unchanged on exit.
  218.  
  219.      xxxx       Array of size at least ( 1 + ( n - 1 )*abs( incx ) ). Before
  220.              entry, the incremented array xxxx must contain the vector x. On
  221.              exit, xxxx is overwritten with the transformed/solution vector x.
  222.  
  223.      iiiinnnnccccxxxx    On entry, iiiinnnnccccxxxx specifies the increment for the elements of xxxx.
  224.              iiiinnnnccccxxxx must not be zero.
  225.              Unchanged on exit.
  226.  
  227.        The matrices  are assumed  to be stored in a  oooonnnneeee ddddiiiimmmmeeeennnnssssiiiioooonnnnaaaallll CCCC aaaarrrrrrrraaaayyyy
  228.        in an analogous fashion as a Fortran array (column major). Therefore,
  229.        the element  A(i+1,j)  of matrix A  is stored  immediately  after the
  230.        element  A(i,j), while  A(i,j+1) is lda  elements apart from  A(i,j).
  231.        The element A(i,j) of the matrix can be accessed directly by reference
  232.        to  a[ (j-1)*lda + (i-1) ].
  233.  
  234.  
  235. AAAAUUUUTTTTHHHHOOOORRRRSSSS
  236.           Jack Dongarra, Argonne National Laboratory.
  237.           Iain Duff, AERE Harwell.
  238.           Jeremy Du Croz, Numerical Algorithms Group Ltd.
  239.           Sven Hammarling, Numerical Algorithms Group Ltd.
  240.  
  241.  
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  252.  
  253.  
  254.  
  255.  
  256.  
  257.  
  258.  
  259.  
  260.  
  261.                                                                         PPPPaaaaggggeeee 4444
  262.  
  263.  
  264.  
  265.