.TH "latps" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME latps \- latps: triangular solve with robust scaling .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBclatps\fP (uplo, trans, diag, normin, n, ap, x, scale, cnorm, info)" .br .RI "\fBCLATPS\fP solves a triangular system of equations with the matrix held in packed storage\&. " .ti -1c .RI "subroutine \fBdlatps\fP (uplo, trans, diag, normin, n, ap, x, scale, cnorm, info)" .br .RI "\fBDLATPS\fP solves a triangular system of equations with the matrix held in packed storage\&. " .ti -1c .RI "subroutine \fBslatps\fP (uplo, trans, diag, normin, n, ap, x, scale, cnorm, info)" .br .RI "\fBSLATPS\fP solves a triangular system of equations with the matrix held in packed storage\&. " .ti -1c .RI "subroutine \fBzlatps\fP (uplo, trans, diag, normin, n, ap, x, scale, cnorm, info)" .br .RI "\fBZLATPS\fP solves a triangular system of equations with the matrix held in packed storage\&. " .in -1c .SH "Detailed Description" .PP .SH "Function Documentation" .PP .SS "subroutine clatps (character uplo, character trans, character diag, character normin, integer n, complex, dimension( * ) ap, complex, dimension( * ) x, real scale, real, dimension( * ) cnorm, integer info)" .PP \fBCLATPS\fP solves a triangular system of equations with the matrix held in packed storage\&. .PP \fBPurpose:\fP .RS 4 .PP .nf CLATPS solves one of the triangular systems A * x = s*b, A**T * x = s*b, or A**H * x = s*b, with scaling to prevent overflow, where A is an upper or lower triangular matrix stored in packed form\&. Here A**T denotes the transpose of A, A**H denotes the conjugate 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 CTPSV 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower triangular\&. = 'U': Upper triangular = 'L': Lower triangular .fi .PP .br \fITRANS\fP .PP .nf TRANS is CHARACTER*1 Specifies the operation applied to A\&. = 'N': Solve A * x = s*b (No transpose) = 'T': Solve A**T * x = s*b (Transpose) = 'C': Solve A**H * x = s*b (Conjugate transpose) .fi .PP .br \fIDIAG\fP .PP .nf DIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular\&. = 'N': Non-unit triangular = 'U': Unit triangular .fi .PP .br \fINORMIN\fP .PP .nf NORMIN is CHARACTER*1 Specifies whether CNORM has been set or not\&. = 'Y': CNORM contains the column norms on entry = 'N': CNORM is not set on entry\&. On exit, the norms will be computed and stored in CNORM\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fIAP\fP .PP .nf AP is COMPLEX 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 = '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\&. .fi .PP .br \fIX\fP .PP .nf X is COMPLEX array, dimension (N) On entry, the right hand side b of the triangular system\&. On exit, X is overwritten by the solution vector x\&. .fi .PP .br \fISCALE\fP .PP .nf SCALE is REAL The scaling factor s for the triangular system A * x = s*b, A**T * x = s*b, or A**H * 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\&. .fi .PP .br \fICNORM\fP .PP .nf CNORM is REAL array, dimension (N) If NORMIN = 'Y', 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 = 'N', CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) must be greater than or equal to the 1-norm\&. If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1-norm of the offdiagonal part of the j-th column of A\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf A rough bound on x is computed; if that is less than overflow, CTPSV is called, otherwise, specific code is used which checks for possible overflow or divide-by-zero at every operation\&. A columnwise scheme is used for solving A*x = b\&. The basic algorithm if A is lower triangular is x[1:n] := b[1:n] for j = 1, \&.\&.\&., n x(j) := x(j) / A(j,j) x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j] end Define bounds on the components of x after j iterations of the loop: M(j) = bound on x[1:j] G(j) = bound on x[j+1:n] Initially, let M(0) = 0 and G(0) = max{x(i), i=1,\&.\&.\&.,n}\&. Then for iteration j+1 we have M(j+1) <= G(j) / | A(j+1,j+1) | G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] | <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | ) where CNORM(j+1) is greater than or equal to the infinity-norm of column j+1 of A, not counting the diagonal\&. Hence G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | ) 1<=i<=j and |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| ) 1<=i< j Since |x(j)| <= M(j), we use the Level 2 BLAS routine CTPSV if the reciprocal of the largest M(j), j=1,\&.\&.,n, is larger than max(underflow, 1/overflow)\&. 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**T *x = b or A**H *x = b\&. The basic algorithm for A upper triangular is for j = 1, \&.\&.\&., n x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j) end We simultaneously compute two bounds G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j M(j) = bound on x(i), 1<=i<=j 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 M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) | <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| ) 1<=i<=j and we can safely call CTPSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow)\&. .fi .PP .RE .PP .SS "subroutine dlatps (character uplo, character trans, character diag, character normin, integer n, double precision, dimension( * ) ap, double precision, dimension( * ) x, double precision scale, double precision, dimension( * ) cnorm, integer info)" .PP \fBDLATPS\fP solves a triangular system of equations with the matrix held in packed storage\&. .PP \fBPurpose:\fP .RS 4 .PP .nf DLATPS solves one of the triangular systems A *x = s*b or A**T*x = s*b with scaling to prevent overflow, where A is an upper or lower triangular matrix stored in packed form\&. Here A**T denotes the 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower triangular\&. = 'U': Upper triangular = 'L': Lower triangular .fi .PP .br \fITRANS\fP .PP .nf TRANS is CHARACTER*1 Specifies the operation applied to A\&. = 'N': Solve A * x = s*b (No transpose) = 'T': Solve A**T* x = s*b (Transpose) = 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose) .fi .PP .br \fIDIAG\fP .PP .nf DIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular\&. = 'N': Non-unit triangular = 'U': Unit triangular .fi .PP .br \fINORMIN\fP .PP .nf NORMIN is CHARACTER*1 Specifies whether CNORM has been set or not\&. = 'Y': CNORM contains the column norms on entry = 'N': CNORM is not set on entry\&. On exit, the norms will be computed and stored in CNORM\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fIAP\fP .PP .nf AP is 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 = '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\&. .fi .PP .br \fIX\fP .PP .nf X is 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\&. .fi .PP .br \fISCALE\fP .PP .nf SCALE is DOUBLE PRECISION The scaling factor s for the triangular system A * x = s*b or A**T* 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\&. .fi .PP .br \fICNORM\fP .PP .nf CNORM is DOUBLE PRECISION array, dimension (N) If NORMIN = 'Y', 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 = 'N', CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) must be greater than or equal to the 1-norm\&. If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1-norm of the offdiagonal part of the j-th column of A\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf 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\&. A columnwise scheme is used for solving A*x = b\&. The basic algorithm if A is lower triangular is x[1:n] := b[1:n] for j = 1, \&.\&.\&., n x(j) := x(j) / A(j,j) x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j] end Define bounds on the components of x after j iterations of the loop: M(j) = bound on x[1:j] G(j) = bound on x[j+1:n] Initially, let M(0) = 0 and G(0) = max{x(i), i=1,\&.\&.\&.,n}\&. Then for iteration j+1 we have M(j+1) <= G(j) / | A(j+1,j+1) | G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] | <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | ) where CNORM(j+1) is greater than or equal to the infinity-norm of column j+1 of A, not counting the diagonal\&. Hence G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | ) 1<=i<=j and |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| ) 1<=i< j 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 max(underflow, 1/overflow)\&. 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**T*x = b\&. The basic algorithm for A upper triangular is for j = 1, \&.\&.\&., n x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j) end We simultaneously compute two bounds G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j M(j) = bound on x(i), 1<=i<=j 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 M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) | <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| ) 1<=i<=j and we can safely call DTPSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow)\&. .fi .PP .RE .PP .SS "subroutine slatps (character uplo, character trans, character diag, character normin, integer n, real, dimension( * ) ap, real, dimension( * ) x, real scale, real, dimension( * ) cnorm, integer info)" .PP \fBSLATPS\fP solves a triangular system of equations with the matrix held in packed storage\&. .PP \fBPurpose:\fP .RS 4 .PP .nf SLATPS solves one of the triangular systems A *x = s*b or A**T*x = s*b with scaling to prevent overflow, where A is an upper or lower triangular matrix stored in packed form\&. Here A**T denotes the 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 STPSV 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower triangular\&. = 'U': Upper triangular = 'L': Lower triangular .fi .PP .br \fITRANS\fP .PP .nf TRANS is CHARACTER*1 Specifies the operation applied to A\&. = 'N': Solve A * x = s*b (No transpose) = 'T': Solve A**T* x = s*b (Transpose) = 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose) .fi .PP .br \fIDIAG\fP .PP .nf DIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular\&. = 'N': Non-unit triangular = 'U': Unit triangular .fi .PP .br \fINORMIN\fP .PP .nf NORMIN is CHARACTER*1 Specifies whether CNORM has been set or not\&. = 'Y': CNORM contains the column norms on entry = 'N': CNORM is not set on entry\&. On exit, the norms will be computed and stored in CNORM\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fIAP\fP .PP .nf AP is REAL 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 = '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\&. .fi .PP .br \fIX\fP .PP .nf X is REAL array, dimension (N) On entry, the right hand side b of the triangular system\&. On exit, X is overwritten by the solution vector x\&. .fi .PP .br \fISCALE\fP .PP .nf SCALE is REAL The scaling factor s for the triangular system A * x = s*b or A**T* 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\&. .fi .PP .br \fICNORM\fP .PP .nf CNORM is REAL array, dimension (N) If NORMIN = 'Y', 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 = 'N', CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) must be greater than or equal to the 1-norm\&. If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1-norm of the offdiagonal part of the j-th column of A\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf A rough bound on x is computed; if that is less than overflow, STPSV is called, otherwise, specific code is used which checks for possible overflow or divide-by-zero at every operation\&. A columnwise scheme is used for solving A*x = b\&. The basic algorithm if A is lower triangular is x[1:n] := b[1:n] for j = 1, \&.\&.\&., n x(j) := x(j) / A(j,j) x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j] end Define bounds on the components of x after j iterations of the loop: M(j) = bound on x[1:j] G(j) = bound on x[j+1:n] Initially, let M(0) = 0 and G(0) = max{x(i), i=1,\&.\&.\&.,n}\&. Then for iteration j+1 we have M(j+1) <= G(j) / | A(j+1,j+1) | G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] | <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | ) where CNORM(j+1) is greater than or equal to the infinity-norm of column j+1 of A, not counting the diagonal\&. Hence G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | ) 1<=i<=j and |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| ) 1<=i< j Since |x(j)| <= M(j), we use the Level 2 BLAS routine STPSV if the reciprocal of the largest M(j), j=1,\&.\&.,n, is larger than max(underflow, 1/overflow)\&. 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**T*x = b\&. The basic algorithm for A upper triangular is for j = 1, \&.\&.\&., n x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j) end We simultaneously compute two bounds G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j M(j) = bound on x(i), 1<=i<=j 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 M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) | <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| ) 1<=i<=j and we can safely call STPSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow)\&. .fi .PP .RE .PP .SS "subroutine zlatps (character uplo, character trans, character diag, character normin, integer n, complex*16, dimension( * ) ap, complex*16, dimension( * ) x, double precision scale, double precision, dimension( * ) cnorm, integer info)" .PP \fBZLATPS\fP solves a triangular system of equations with the matrix held in packed storage\&. .PP \fBPurpose:\fP .RS 4 .PP .nf ZLATPS solves one of the triangular systems A * x = s*b, A**T * x = s*b, or A**H * x = s*b, with scaling to prevent overflow, where A is an upper or lower triangular matrix stored in packed form\&. Here A**T denotes the transpose of A, A**H denotes the conjugate 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 ZTPSV 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower triangular\&. = 'U': Upper triangular = 'L': Lower triangular .fi .PP .br \fITRANS\fP .PP .nf TRANS is CHARACTER*1 Specifies the operation applied to A\&. = 'N': Solve A * x = s*b (No transpose) = 'T': Solve A**T * x = s*b (Transpose) = 'C': Solve A**H * x = s*b (Conjugate transpose) .fi .PP .br \fIDIAG\fP .PP .nf DIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular\&. = 'N': Non-unit triangular = 'U': Unit triangular .fi .PP .br \fINORMIN\fP .PP .nf NORMIN is CHARACTER*1 Specifies whether CNORM has been set or not\&. = 'Y': CNORM contains the column norms on entry = 'N': CNORM is not set on entry\&. On exit, the norms will be computed and stored in CNORM\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fIAP\fP .PP .nf AP is COMPLEX*16 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 = '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\&. .fi .PP .br \fIX\fP .PP .nf X is COMPLEX*16 array, dimension (N) On entry, the right hand side b of the triangular system\&. On exit, X is overwritten by the solution vector x\&. .fi .PP .br \fISCALE\fP .PP .nf SCALE is DOUBLE PRECISION The scaling factor s for the triangular system A * x = s*b, A**T * x = s*b, or A**H * 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\&. .fi .PP .br \fICNORM\fP .PP .nf CNORM is DOUBLE PRECISION array, dimension (N) If NORMIN = 'Y', 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 = 'N', CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) must be greater than or equal to the 1-norm\&. If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1-norm of the offdiagonal part of the j-th column of A\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf A rough bound on x is computed; if that is less than overflow, ZTPSV is called, otherwise, specific code is used which checks for possible overflow or divide-by-zero at every operation\&. A columnwise scheme is used for solving A*x = b\&. The basic algorithm if A is lower triangular is x[1:n] := b[1:n] for j = 1, \&.\&.\&., n x(j) := x(j) / A(j,j) x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j] end Define bounds on the components of x after j iterations of the loop: M(j) = bound on x[1:j] G(j) = bound on x[j+1:n] Initially, let M(0) = 0 and G(0) = max{x(i), i=1,\&.\&.\&.,n}\&. Then for iteration j+1 we have M(j+1) <= G(j) / | A(j+1,j+1) | G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] | <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | ) where CNORM(j+1) is greater than or equal to the infinity-norm of column j+1 of A, not counting the diagonal\&. Hence G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | ) 1<=i<=j and |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| ) 1<=i< j Since |x(j)| <= M(j), we use the Level 2 BLAS routine ZTPSV if the reciprocal of the largest M(j), j=1,\&.\&.,n, is larger than max(underflow, 1/overflow)\&. 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**T *x = b or A**H *x = b\&. The basic algorithm for A upper triangular is for j = 1, \&.\&.\&., n x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j) end We simultaneously compute two bounds G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j M(j) = bound on x(i), 1<=i<=j 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 M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) | <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| ) 1<=i<=j and we can safely call ZTPSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow)\&. .fi .PP .RE .PP .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.