SSPTRF (3lapack)

compute the factorization of a real symmetric matrix A stored in packed format using the Bunch-Kaufman diagonal pivoting method

SYNOPSIS

  SUBROUTINE SSPTRF( UPLO, N, AP, IPIV, INFO )

      CHARACTER      UPLO

      INTEGER        INFO, N

      INTEGER        IPIV( * )

      REAL           AP( * )

PURPOSE

  SSPTRF computes the factorization of a real symmetric matrix A stored in
  packed format using the Bunch-Kaufman diagonal pivoting method:

     A = U*D*U**T  or  A = L*D*L**T

  where U (or L) is a product of permutation and unit upper (lower)
  triangular matrices, and D is symmetric and block diagonal with 1-by-1 and
  2-by-2 diagonal blocks.

ARGUMENTS

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  AP      (input/output) REAL array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix A,
          packed columnwise in a linear array.  The j-th column of A is
          stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2)
          = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) =
          A(i,j) for j<=i<=n.

          On exit, the block diagonal matrix D and the multipliers used to
          obtain the factor U or L, stored as a packed triangular matrix
          overwriting A (see below for further details).

  IPIV    (output) INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D.  If
          IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged
          and D(k,k) is a 1-by-1 diagonal block.  If UPLO = 'U' and IPIV(k) =
          IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were
          interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block.  If
          UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1
          and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2
          diagonal block.

  INFO    (output) INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, D(i,i) is exactly zero.  The factorization has
          been completed, but the block diagonal matrix D is exactly
          singular, and division by zero will occur if it is used to solve a
          system of equations.

FURTHER DETAILS

  If UPLO = 'U', then A = U*D*U', where
     U = P(n)*U(n)* ... *P(k)U(k)* ...,
  i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in
  steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2
  diagonal blocks D(k).  P(k) is a permutation matrix as defined by IPIV(k),
  and U(k) is a unit upper triangular matrix, such that if the diagonal block
  D(k) is of order s (s = 1 or 2), then

             (   I    v    0   )   k-s
     U(k) =  (   0    I    0   )   s
             (   0    0    I   )   n-k
                k-s   s   n-k

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).  If s = 2,
  the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and
  v overwrites A(1:k-2,k-1:k).

  If UPLO = 'L', then A = L*D*L', where
     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in
  steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2
  diagonal blocks D(k).  P(k) is a permutation matrix as defined by IPIV(k),
  and L(k) is a unit lower triangular matrix, such that if the diagonal block
  D(k) is of order s (s = 1 or 2), then

             (   I    0     0   )  k-1
     L(k) =  (   0    I     0   )  s
             (   0    v     I   )  n-k-s+1
                k-1   s  n-k-s+1

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).  If s = 2,
  the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and
  v overwrites A(k+2:n,k:k+1).