.TH DLATPS 1 "November 2006" " LAPACK auxiliary routine (version 3.1) " " LAPACK auxiliary routine (version 3.1) " .SH NAME DLATPS - one of the triangular systems A *x = s*b or A\(aq*x = s*b with scaling to prevent overflow, where A is an upper or lower triangular matrix stored in packed form .SH SYNOPSIS .TP 19 SUBROUTINE DLATPS( UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE, CNORM, INFO ) .TP 19 .ti +4 CHARACTER DIAG, NORMIN, TRANS, UPLO .TP 19 .ti +4 INTEGER INFO, N .TP 19 .ti +4 DOUBLE PRECISION SCALE .TP 19 .ti +4 DOUBLE PRECISION AP( * ), CNORM( * ), X( * ) .SH PURPOSE DLATPS solves one of the triangular systems transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine DTPSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned. .SH ARGUMENTS .TP 8 UPLO (input) CHARACTER*1 Specifies whether the matrix A is upper or lower triangular. = \(aqU\(aq: Upper triangular .br = \(aqL\(aq: Lower triangular .TP 8 TRANS (input) CHARACTER*1 Specifies the operation applied to A. = \(aqN\(aq: Solve A * x = s*b (No transpose) .br = \(aqT\(aq: Solve A\(aq* x = s*b (Transpose) .br = \(aqC\(aq: Solve A\(aq* x = s*b (Conjugate transpose = Transpose) .TP 8 DIAG (input) CHARACTER*1 Specifies whether or not the matrix A is unit triangular. = \(aqN\(aq: Non-unit triangular .br = \(aqU\(aq: Unit triangular .TP 8 NORMIN (input) CHARACTER*1 Specifies whether CNORM has been set or not. = \(aqY\(aq: CNORM contains the column norms on entry .br = \(aqN\(aq: CNORM is not set on entry. On exit, the norms will be computed and stored in CNORM. .TP 8 N (input) INTEGER The order of the matrix A. N >= 0. .TP 8 AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) The upper or lower triangular matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = \(aqU\(aq, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = \(aqL\(aq, AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. .TP 8 X (input/output) DOUBLE PRECISION array, dimension (N) On entry, the right hand side b of the triangular system. On exit, X is overwritten by the solution vector x. .TP 8 SCALE (output) DOUBLE PRECISION The scaling factor s for the triangular system A * x = s*b or A\(aq* x = s*b. If SCALE = 0, the matrix A is singular or badly scaled, and the vector x is an exact or approximate solution to A*x = 0. .TP 8 CNORM (input or output) DOUBLE PRECISION array, dimension (N) If NORMIN = \(aqY\(aq, CNORM is an input argument and CNORM(j) contains the norm of the off-diagonal part of the j-th column of A. If TRANS = \(aqN\(aq, CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = \(aqT\(aq or \(aqC\(aq, CNORM(j) must be greater than or equal to the 1-norm. If NORMIN = \(aqN\(aq, CNORM is an output argument and CNORM(j) returns the 1-norm of the offdiagonal part of the j-th column of A. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -k, the k-th argument had an illegal value .SH FURTHER DETAILS A rough bound on x is computed; if that is less than overflow, DTPSV is called, otherwise, specific code is used which checks for possible overflow or divide-by-zero at every operation. .br A columnwise scheme is used for solving A*x = b. The basic algorithm if A is lower triangular is .br x[1:n] := b[1:n] .br for j = 1, ..., n .br x(j) := x(j) / A(j,j) .br x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j] .br end .br Define bounds on the components of x after j iterations of the loop: M(j) = bound on x[1:j] .br G(j) = bound on x[j+1:n] .br Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}. .br Then for iteration j+1 we have .br M(j+1) <= G(j) / | A(j+1,j+1) | .br G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] | .br <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | ) .br where CNORM(j+1) is greater than or equal to the infinity-norm of column j+1 of A, not counting the diagonal. Hence .br G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | ) .br 1<=i<=j .br and .br |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| ) 1<=i< j .br Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTPSV if the reciprocal of the largest M(j), j=1,..,n, is larger than .br max(underflow, 1/overflow). .br The bound on x(j) is also used to determine when a step in the columnwise method can be performed without fear of overflow. If the computed bound is greater than a large constant, x is scaled to prevent overflow, but if the bound overflows, x is set to 0, x(j) to 1, and scale to 0, and a non-trivial solution to A*x = 0 is found. Similarly, a row-wise scheme is used to solve A\(aq*x = b. The basic algorithm for A upper triangular is .br for j = 1, ..., n .br x(j) := ( b(j) - A[1:j-1,j]\(aq * x[1:j-1] ) / A(j,j) end .br We simultaneously compute two bounds .br G(j) = bound on ( b(i) - A[1:i-1,i]\(aq * x[1:i-1] ), 1<=i<=j M(j) = bound on x(i), 1<=i<=j .br The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1. Then the bound on x(j) is .br M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) | .br <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| ) 1<=i<=j .br and we can safely call DTPSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow). .br