Purpose
To compute P = H*X or P = X*H, where H is an upper Hessenberg matrix and X is a symmetric matrix.Specification
      SUBROUTINE MB01OS( UPLO, TRANS, N, H, LDH, X, LDX, P, LDP, INFO )
C     .. Scalar Arguments ..
      CHARACTER         TRANS, UPLO
      INTEGER           INFO, LDH, LDP, LDX, N
C     .. Array Arguments ..
      DOUBLE PRECISION  H(LDH,*), P(LDP,*), X(LDX,*)
Arguments
Mode Parameters
  UPLO    CHARACTER*1
          Specifies which triangle of the symmetric matrix X is
          given as follows:
          = 'U':  the upper triangular part is given;
          = 'L':  the lower triangular part is given.
  TRANS   CHARACTER*1
          Specifies the operation to be performed as follows:
          = 'N':         compute P = H*X;
          = 'T' or 'C':  compute P = X*H.
Input/Output Parameters
  N       (input) INTEGER
          The order of the matrices H, X, and P.  N >= 0.
  H       (input) DOUBLE PRECISION array, dimension (LDH,N)
          On entry, the leading N-by-N upper Hessenberg part of this
          array must contain the upper Hessenberg matrix H.
          The remaining part of this array is not referenced.
  LDH     INTEGER
          The leading dimension of the array H.  LDH >= MAX(1,N).
  X       (input) DOUBLE PRECISION array, dimension (LDX,N)
          On entry, if UPLO = 'U', the leading N-by-N upper
          triangular part of this array must contain the upper
          triangular part of the symmetric matrix X and the strictly
          lower triangular part of the array is not referenced.
          On entry, if UPLO = 'L', the leading N-by-N lower
          triangular part of this array must contain the lower
          triangular part of the symmetric matrix X and the strictly
          upper triangular part of the array is not referenced.
  LDX     INTEGER
          The leading dimension of the array X.  LDX >= MAX(1,N).
  P       (output) DOUBLE PRECISION array, dimension (LDP,N)
          On exit, the leading N-by-N part of this array contains
          the computed matrix P.
  LDP     INTEGER
          The leading dimension of the array P.  LDP >= MAX(1,N).
Error Indicator
  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -k, the k-th argument had an illegal
                value.
Method
The matrix expression is efficiently evaluated taking the structure into account, and using inline code and BLAS routines. Let X = U + sL, where U is upper triangular and sL is strictly lower triangular. Then, P = H*X = H*U + H*sL = H*U + H*sU', where sU is the strictly upper triangular part of X. Similarly, P = X*H = L'*H + sL*H, where L is lower triangular, and X = L + sL'. Note that H*U and L'*H are both upper Hessenberg. However, when UPLO = 'L' and TRANS = 'N', or when UPLO = 'U' and TRANS = 'T', then the matrix P is full. The computations are done similarly.Numerical Aspects
The algorithm requires approximately N**3/2 operations.Further Comments
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