**Purpose**

To solve a set of systems of linear equations, A'*A*X = B, or, in the implicit form, f(A)*X = B, with A'*A or f(A) positive definite, using symmetric Gaussian elimination.

SUBROUTINE MB02XD( FORM, STOR, UPLO, F, M, N, NRHS, IPAR, LIPAR, $ DPAR, LDPAR, A, LDA, B, LDB, ATA, LDATA, DWORK, $ LDWORK, INFO ) C .. Scalar Arguments .. CHARACTER FORM, STOR, UPLO INTEGER INFO, LDA, LDATA, LDB, LDPAR, LDWORK, LIPAR, M, $ N, NRHS C .. Array Arguments .. DOUBLE PRECISION A(LDA,*), ATA(*), B(LDB,*), DPAR(*), DWORK(*) INTEGER IPAR(*)

**Mode Parameters**

FORM CHARACTER*1 Specifies the form in which the matrix A is provided, as follows: = 'S' : standard form, the matrix A is given; = 'F' : the implicit, function form f(A) is provided. If FORM = 'F', then the routine F is called to compute the matrix A'*A. STOR CHARACTER*1 Specifies the storage scheme for the symmetric matrix A'*A, as follows: = 'F' : full storage is used; = 'P' : packed storage is used. UPLO CHARACTER*1 Specifies which part of the matrix A'*A is stored, as follows: = 'U' : the upper triagular part is stored; = 'L' : the lower triagular part is stored.

F EXTERNAL If FORM = 'F', then F is a subroutine which calculates the value of f(A) = A'*A, for given A. If FORM = 'S', then F is not called. F must have the following interface: SUBROUTINE F( STOR, UPLO, N, IPAR, LIPAR, DPAR, LDPAR, A, $ LDA, ATA, LDATA, DWORK, LDWORK, INFO ) where STOR (input) CHARACTER*1 Specifies the storage scheme for the symmetric matrix A'*A, as follows: = 'F' : full storage is used; = 'P' : packed storage is used. UPLO (input) CHARACTER*1 Specifies which part of the matrix A'*A is stored, as follows: = 'U' : the upper triagular part is stored; = 'L' : the lower triagular part is stored. N (input) INTEGER The order of the matrix A'*A. N >= 0. IPAR (input) INTEGER array, dimension (LIPAR) The integer parameters describing the structure of the matrix A. LIPAR (input) INTEGER The length of the array IPAR. LIPAR >= 0. DPAR (input) DOUBLE PRECISION array, dimension (LDPAR) The real parameters needed for solving the problem. LDPAR (input) INTEGER The length of the array DPAR. LDPAR >= 0. A (input) DOUBLE PRECISION array, dimension (LDA, NC), where NC is the number of columns. The leading NR-by-NC part of this array must contain the (compressed) representation of the matrix A, where NR is the number of rows of A (function of IPAR entries). LDA (input) INTEGER The leading dimension of the array A. LDA >= MAX(1,NR). ATA (output) DOUBLE PRECISION array, dimension (LDATA,N), if STOR = 'F', dimension (N*(N+1)/2), if STOR = 'P'. The leading N-by-N (if STOR = 'F'), or N*(N+1)/2 (if STOR = 'P') part of this array contains the upper or lower triangle of the matrix A'*A, depending on UPLO = 'U', or UPLO = 'L', respectively, stored either as a two-dimensional, or one-dimensional array, depending on STOR. LDATA (input) INTEGER The leading dimension of the array ATA. LDATA >= MAX(1,N), if STOR = 'F'. LDATA >= 1, if STOR = 'P'. DWORK DOUBLE PRECISION array, dimension (LDWORK) The workspace array for subroutine F. LDWORK (input) INTEGER The size of the array DWORK (as large as needed in the subroutine F). INFO INTEGER Error indicator, set to a negative value if an input scalar argument is erroneous, and to positive values for other possible errors in the subroutine F. The LAPACK Library routine XERBLA should be used in conjunction with negative INFO. INFO must be zero if the subroutine finished successfully. Parameters marked with "(input)" must not be changed.

M (input) INTEGER The number of rows of the matrix A. M >= 0. N (input) INTEGER The order of the matrix A'*A, the number of columns of the matrix A, and the number of rows of the matrix X. N >= 0. NRHS (input) INTEGER The number of columns of the matrices B and X. NRHS >= 0. IPAR (input) INTEGER array, dimension (LIPAR) If FORM = 'F', the integer parameters describing the structure of the matrix A. This parameter is ignored if FORM = 'S'. LIPAR (input) INTEGER The length of the array IPAR. LIPAR >= 0. DPAR (input) DOUBLE PRECISION array, dimension (LDPAR) If FORM = 'F', the real parameters needed for solving the problem. This parameter is ignored if FORM = 'S'. LDPAR (input) INTEGER The length of the array DPAR. LDPAR >= 0. A (input) DOUBLE PRECISION array, dimension (LDA, N), if FORM = 'S', dimension (LDA, NC), if FORM = 'F', where NC is the number of columns. If FORM = 'S', the leading M-by-N part of this array must contain the matrix A. If FORM = 'F', the leading NR-by-NC part of this array must contain an appropriate representation of matrix A, where NR is the number of rows. If FORM = 'F', this array is not referenced by this routine itself, except in the call to the routine F. LDA INTEGER The leading dimension of array A. LDA >= MAX(1,M), if FORM = 'S'; LDA >= MAX(1,NR), if FORM = 'F'. B (input/output) DOUBLE PRECISION array, dimension (LDB, NRHS) On entry, the leading N-by-NRHS part of this array must contain the right hand side matrix B. On exit, if INFO = 0 and M (or NR) is nonzero, the leading N-by-NRHS part of this array contains the solution X of the set of systems of linear equations A'*A*X = B or f(A)*X = B. If M (or NR) is zero, then B is unchanged. LDB INTEGER The leading dimension of array B. LDB >= MAX(1,N). ATA (output) DOUBLE PRECISION array, dimension (LDATA,N), if STOR = 'F', dimension (N*(N+1)/2), if STOR = 'P'. The leading N-by-N (if STOR = 'F'), or N*(N+1)/2 (if STOR = 'P') part of this array contains the upper or lower triangular Cholesky factor of the matrix A'*A, depending on UPLO = 'U', or UPLO = 'L', respectively, stored either as a two-dimensional, or one-dimensional array, depending on STOR. LDATA INTEGER The leading dimension of the array ATA. LDATA >= MAX(1,N), if STOR = 'F'. LDATA >= 1, if STOR = 'P'.

DWORK DOUBLE PRECISION array, dimension (LDWORK) LDWORK INTEGER The length of the array DWORK.

INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value; > 0: if INFO = i, then the (i,i) element of the triangular factor of the matrix A'*A is exactly zero (the matrix A'*A is exactly singular); if INFO = j > n, then F returned with INFO = j-n.

The matrix A'*A is built either directly (if FORM = 'S'), or implicitly, by calling the routine F. Then, A'*A is Cholesky factored and its factor is used to solve the set of systems of linear equations, A'*A*X = B.

[1] Golub, G.H. and van Loan, C.F. Matrix Computations. Third Edition. M. D. Johns Hopkins University Press, Baltimore, 1996. [2] Anderson, E., Bai, Z., Bischof, C., Blackford, Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D. LAPACK Users' Guide: Third Edition, SIAM, Philadelphia, 1999.

For speed, this routine does not check for near singularity of the matrix A'*A. If the matrix A is nearly rank deficient, then the computed X could be inaccurate. Estimates of the reciprocal condition numbers of the matrices A and A'*A can be obtained using LAPACK routines DGECON and DPOCON (DPPCON), respectively. The approximate number of floating point operations is (M+3)*N**2/2 + N**3/6 + NRHS*N**2, if FORM = 'S', f + N**3/6 + NRHS*N**2, if FORM = 'F', where M is the number of rows of A, and f is the number of floating point operations required by the subroutine F.

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**Program Text**

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