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NAME DPTSVX - use the factorization A = L*D*L**T to compute the solution to a real system of linear equations A*X = B, where A is an N-by-N symmetric positive definite tridiagonal matrix and X and B are N-by-NRHS matrices SYNOPSIS SUBROUTINE DPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, INFO ) CHARACTER FACT INTEGER INFO, LDB, LDX, N, NRHS DOUBLE PRECISION RCOND DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ), E( * ), EF( * ), FERR( * ), WORK( * ), X( LDX, * ) PURPOSE DPTSVX uses the factorization A = L*D*L**T to compute the solution to a real system of linear equations A*X = B, where A is an N-by-N symmetric positive definite tridiagonal matrix and X and B are N-by-NRHS matrices. Error bounds on the solution and a condition estimate are also provided. DESCRIPTION The following steps are performed: 1. If FACT = 'N', the matrix A is factored as A = L*D*L**T, where L is a unit lower bidiagonal matrix and D is diagonal. The factorization can also be regarded as having the form A = U**T*D*U. 2. The factored form of A is used to compute the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, steps 3 and 4 are skipped. 3. The system of equations is solved for X using the fac- tored form of A. 4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. ARGUMENTS FACT (input) CHARACTER*1 Specifies whether or not the factored form of A has been supplied on entry. = 'F': On entry, DF and EF contain the factored form of A. D, E, DF, and EF will not be modified. = 'N': The matrix A will be copied to DF and EF and factored. N (input) INTEGER The order of the matrix A. N >= 0. NRHS (input) INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. D (input) DOUBLE PRECISION array, dimension (N) The n diagonal elements of the tridiagonal matrix A. E (input) DOUBLE PRECISION array, dimension (N-1) The (n-1) subdiagonal elements of the tridiagonal matrix A. DF (input or output) DOUBLE PRECISION array, dimension (N) If FACT = 'F', then DF is an input argument and on entry contains the n diagonal elements of the diago- nal matrix D from the L*D*L**T factorization of A. If FACT = 'N', then DF is an output argument and on exit contains the n diagonal elements of the diago- nal matrix D from the L*D*L**T factorization of A. EF (input or output) DOUBLE PRECISION array, dimension (N- 1) If FACT = 'F', then EF is an input argument and on entry contains the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**T factori- zation of A. If FACT = 'N', then EF is an output argument and on exit contains the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**T factorization of A. B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) The N-by-NRHS right hand side matrix B. LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) If INFO = 0, the N-by-NRHS solution matrix X. LDX (input) INTEGER The leading dimension of the array X. LDX >= max(1,N). RCOND (output) DOUBLE PRECISION The reciprocal condition number of the matrix A. If RCOND is less than the machine precision (in partic- ular, if RCOND = 0), the matrix is singular to work- ing precision. This condition is indicated by a return code of INFO > 0, and the solution and error bounds are not computed. FERR (output) DOUBLE PRECISION array, dimension (NRHS) The estimated forward error bounds for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), FERR(j) bounds the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). BERR (output) DOUBLE PRECISION array, dimension (NRHS) The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution). WORK (workspace) DOUBLE PRECISION array, dimension (2*N) INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, and i is <= N the leading minor of order i of A is not positive definite, so the factorization could not be completed unless i = N, and the solution and error bounds could not be com- puted. = N+1 RCOND is less than machine precision. The factorization has been completed, but the matrix is singular to working precision, and the solution and error bounds have not been computed.