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.

UPLO

          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

AP

          AP is 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

          IPIV is 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

          INFO is 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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2011

  5-96 - Based on modifications by J. Lewis, Boeing Computer Services


         Company


  If UPLO = 'U', then A = U*D*U**T, where


     U = P(n)*U(n)* ... <em>P(k)U(k)</em> ...,


  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**T, where


     L = P(1)*L(1)* ... <em>P(k)*L(k)</em> ...,


  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).