--- name: "hprfs" section: "3" platform: "linux" package: "lapack-doc" version: "3.12.1-2" release: "Arch Linux 2026-11-01" platforms: ["linux"] url: "https://new.carta.tech/hprfs" --- # hprfs(3) {hp,sp}rfs: iterative refinement ## NAME hprfs - {hp,sp}rfs: iterative refinement ## SYNOPSIS ### Functions subroutine **chprfs** (uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)\ **CHPRFS** subroutine **csprfs** (uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)\ **CSPRFS** subroutine **dsprfs** (uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info)\ **DSPRFS** subroutine **ssprfs** (uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info)\ **SSPRFS** subroutine **zhprfs** (uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)\ **ZHPRFS** subroutine **zsprfs** (uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)\ **ZSPRFS** ## Detailed Description ## Function Documentation ### subroutine chprfs (character uplo, integer n, integer nrhs, complex, dimension( \* ) ap, complex, dimension( \* ) afp, integer, dimension( \* ) ipiv, complex, dimension( ldb, \* ) b, integer ldb, complex, dimension( ldx, \* ) x, integer ldx, real, dimension( \* ) ferr, real, dimension( \* ) berr, complex, dimension( \* ) work, real, dimension( \* ) rwork, integer info) **CHPRFS** **Purpose:** ``` !> !> CHPRFS improves the computed solution to a system of linear !> equations when the coefficient matrix is Hermitian indefinite !> and packed, and provides error bounds and backward error estimates !> for the solution. !> ``` **Parameters** *UPLO* ``` !> UPLO is CHARACTER*1 !> = 'U': Upper triangle of A is stored; !> = 'L': Lower triangle of A is stored. !> ``` *N* ``` !> N is INTEGER !> The order of the matrix A. N >= 0. !> ``` *NRHS* ``` !> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrices B and X. NRHS >= 0. !> ``` *AP* ``` !> AP is COMPLEX array, dimension (N*(N+1)/2) !> The upper or lower triangle of the Hermitian 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)*(2*n-j)/2) = A(i,j) for j<=i<=n. !> ``` *AFP* ``` !> AFP is COMPLEX array, dimension (N*(N+1)/2) !> The factored form of the matrix A. AFP contains the block !> diagonal matrix D and the multipliers used to obtain the !> factor U or L from the factorization A = U*D*U**H or !> A = L*D*L**H as computed by CHPTRF, stored as a packed !> triangular matrix. !> ``` *IPIV* ``` !> IPIV is INTEGER array, dimension (N) !> Details of the interchanges and the block structure of D !> as determined by CHPTRF. !> ``` *B* ``` !> B is COMPLEX array, dimension (LDB,NRHS) !> The right hand side matrix B. !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,N). !> ``` *X* ``` !> X is COMPLEX array, dimension (LDX,NRHS) !> On entry, the solution matrix X, as computed by CHPTRS. !> On exit, the improved solution matrix X. !> ``` *LDX* ``` !> LDX is INTEGER !> The leading dimension of the array X. LDX >= max(1,N). !> ``` *FERR* ``` !> FERR is REAL array, dimension (NRHS) !> The estimated forward error bound 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) !> is an estimated upper bound for the magnitude of the largest !> element in (X(j) - XTRUE) divided by the magnitude of the !> largest element in X(j). The estimate is as reliable as !> the estimate for RCOND, and is almost always a slight !> overestimate of the true error. !> ``` *BERR* ``` !> BERR is REAL 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* ``` !> WORK is COMPLEX array, dimension (2*N) !> ``` *RWORK* ``` !> RWORK is REAL array, dimension (N) !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> ``` **Internal Parameters:** ``` !> ITMAX is the maximum number of steps of iterative refinement. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **178** of file **chprfs.f**. ### subroutine csprfs (character uplo, integer n, integer nrhs, complex, dimension( \* ) ap, complex, dimension( \* ) afp, integer, dimension( \* ) ipiv, complex, dimension( ldb, \* ) b, integer ldb, complex, dimension( ldx, \* ) x, integer ldx, real, dimension( \* ) ferr, real, dimension( \* ) berr, complex, dimension( \* ) work, real, dimension( \* ) rwork, integer info) **CSPRFS** **Purpose:** ``` !> !> CSPRFS improves the computed solution to a system of linear !> equations when the coefficient matrix is symmetric indefinite !> and packed, and provides error bounds and backward error estimates !> for the solution. !> ``` **Parameters** *UPLO* ``` !> UPLO is CHARACTER*1 !> = 'U': Upper triangle of A is stored; !> = 'L': Lower triangle of A is stored. !> ``` *N* ``` !> N is INTEGER !> The order of the matrix A. N >= 0. !> ``` *NRHS* ``` !> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrices B and X. NRHS >= 0. !> ``` *AP* ``` !> AP is COMPLEX array, dimension (N*(N+1)/2) !> The upper or lower triangle of the symmetric 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)*(2*n-j)/2) = A(i,j) for j<=i<=n. !> ``` *AFP* ``` !> AFP is COMPLEX array, dimension (N*(N+1)/2) !> The factored form of the matrix A. AFP contains the block !> diagonal matrix D and the multipliers used to obtain the !> factor U or L from the factorization A = U*D*U**T or !> A = L*D*L**T as computed by CSPTRF, stored as a packed !> triangular matrix. !> ``` *IPIV* ``` !> IPIV is INTEGER array, dimension (N) !> Details of the interchanges and the block structure of D !> as determined by CSPTRF. !> ``` *B* ``` !> B is COMPLEX array, dimension (LDB,NRHS) !> The right hand side matrix B. !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,N). !> ``` *X* ``` !> X is COMPLEX array, dimension (LDX,NRHS) !> On entry, the solution matrix X, as computed by CSPTRS. !> On exit, the improved solution matrix X. !> ``` *LDX* ``` !> LDX is INTEGER !> The leading dimension of the array X. LDX >= max(1,N). !> ``` *FERR* ``` !> FERR is REAL array, dimension (NRHS) !> The estimated forward error bound 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) !> is an estimated upper bound for the magnitude of the largest !> element in (X(j) - XTRUE) divided by the magnitude of the !> largest element in X(j). The estimate is as reliable as !> the estimate for RCOND, and is almost always a slight !> overestimate of the true error. !> ``` *BERR* ``` !> BERR is REAL 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* ``` !> WORK is COMPLEX array, dimension (2*N) !> ``` *RWORK* ``` !> RWORK is REAL array, dimension (N) !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> ``` **Internal Parameters:** ``` !> ITMAX is the maximum number of steps of iterative refinement. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **178** of file **csprfs.f**. ### subroutine dsprfs (character uplo, integer n, integer nrhs, double precision, dimension( \* ) ap, double precision, dimension( \* ) afp, integer, dimension( \* ) ipiv, double precision, dimension( ldb, \* ) b, integer ldb, double precision, dimension( ldx, \* ) x, integer ldx, double precision, dimension( \* ) ferr, double precision, dimension( \* ) berr, double precision, dimension( \* ) work, integer, dimension( \* ) iwork, integer info) **DSPRFS** **Purpose:** ``` !> !> DSPRFS improves the computed solution to a system of linear !> equations when the coefficient matrix is symmetric indefinite !> and packed, and provides error bounds and backward error estimates !> for the solution. !> ``` **Parameters** *UPLO* ``` !> UPLO is CHARACTER*1 !> = 'U': Upper triangle of A is stored; !> = 'L': Lower triangle of A is stored. !> ``` *N* ``` !> N is INTEGER !> The order of the matrix A. N >= 0. !> ``` *NRHS* ``` !> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrices B and X. NRHS >= 0. !> ``` *AP* ``` !> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2) !> The upper or lower triangle of the symmetric 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)*(2*n-j)/2) = A(i,j) for j<=i<=n. !> ``` *AFP* ``` !> AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2) !> The factored form of the matrix A. AFP contains the block !> diagonal matrix D and the multipliers used to obtain the !> factor U or L from the factorization A = U*D*U**T or !> A = L*D*L**T as computed by DSPTRF, stored as a packed !> triangular matrix. !> ``` *IPIV* ``` !> IPIV is INTEGER array, dimension (N) !> Details of the interchanges and the block structure of D !> as determined by DSPTRF. !> ``` *B* ``` !> B is DOUBLE PRECISION array, dimension (LDB,NRHS) !> The right hand side matrix B. !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,N). !> ``` *X* ``` !> X is DOUBLE PRECISION array, dimension (LDX,NRHS) !> On entry, the solution matrix X, as computed by DSPTRS. !> On exit, the improved solution matrix X. !> ``` *LDX* ``` !> LDX is INTEGER !> The leading dimension of the array X. LDX >= max(1,N). !> ``` *FERR* ``` !> FERR is DOUBLE PRECISION array, dimension (NRHS) !> The estimated forward error bound 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) !> is an estimated upper bound for the magnitude of the largest !> element in (X(j) - XTRUE) divided by the magnitude of the !> largest element in X(j). The estimate is as reliable as !> the estimate for RCOND, and is almost always a slight !> overestimate of the true error. !> ``` *BERR* ``` !> BERR is 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* ``` !> WORK is DOUBLE PRECISION array, dimension (3*N) !> ``` *IWORK* ``` !> IWORK is INTEGER array, dimension (N) !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> ``` **Internal Parameters:** ``` !> ITMAX is the maximum number of steps of iterative refinement. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **177** of file **dsprfs.f**. ### subroutine ssprfs (character uplo, integer n, integer nrhs, real, dimension( \* ) ap, real, dimension( \* ) afp, integer, dimension( \* ) ipiv, real, dimension( ldb, \* ) b, integer ldb, real, dimension( ldx, \* ) x, integer ldx, real, dimension( \* ) ferr, real, dimension( \* ) berr, real, dimension( \* ) work, integer, dimension( \* ) iwork, integer info) **SSPRFS** **Purpose:** ``` !> !> SSPRFS improves the computed solution to a system of linear !> equations when the coefficient matrix is symmetric indefinite !> and packed, and provides error bounds and backward error estimates !> for the solution. !> ``` **Parameters** *UPLO* ``` !> UPLO is CHARACTER*1 !> = 'U': Upper triangle of A is stored; !> = 'L': Lower triangle of A is stored. !> ``` *N* ``` !> N is INTEGER !> The order of the matrix A. N >= 0. !> ``` *NRHS* ``` !> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrices B and X. NRHS >= 0. !> ``` *AP* ``` !> AP is REAL array, dimension (N*(N+1)/2) !> The upper or lower triangle of the symmetric 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)*(2*n-j)/2) = A(i,j) for j<=i<=n. !> ``` *AFP* ``` !> AFP is REAL array, dimension (N*(N+1)/2) !> The factored form of the matrix A. AFP contains the block !> diagonal matrix D and the multipliers used to obtain the !> factor U or L from the factorization A = U*D*U**T or !> A = L*D*L**T as computed by SSPTRF, stored as a packed !> triangular matrix. !> ``` *IPIV* ``` !> IPIV is INTEGER array, dimension (N) !> Details of the interchanges and the block structure of D !> as determined by SSPTRF. !> ``` *B* ``` !> B is REAL array, dimension (LDB,NRHS) !> The right hand side matrix B. !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,N). !> ``` *X* ``` !> X is REAL array, dimension (LDX,NRHS) !> On entry, the solution matrix X, as computed by SSPTRS. !> On exit, the improved solution matrix X. !> ``` *LDX* ``` !> LDX is INTEGER !> The leading dimension of the array X. LDX >= max(1,N). !> ``` *FERR* ``` !> FERR is REAL array, dimension (NRHS) !> The estimated forward error bound 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) !> is an estimated upper bound for the magnitude of the largest !> element in (X(j) - XTRUE) divided by the magnitude of the !> largest element in X(j). The estimate is as reliable as !> the estimate for RCOND, and is almost always a slight !> overestimate of the true error. !> ``` *BERR* ``` !> BERR is REAL 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* ``` !> WORK is REAL array, dimension (3*N) !> ``` *IWORK* ``` !> IWORK is INTEGER array, dimension (N) !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> ``` **Internal Parameters:** ``` !> ITMAX is the maximum number of steps of iterative refinement. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **177** of file **ssprfs.f**. ### subroutine zhprfs (character uplo, integer n, integer nrhs, complex\*16, dimension( \* ) ap, complex\*16, dimension( \* ) afp, integer, dimension( \* ) ipiv, complex\*16, dimension( ldb, \* ) b, integer ldb, complex\*16, dimension( ldx, \* ) x, integer ldx, double precision, dimension( \* ) ferr, double precision, dimension( \* ) berr, complex\*16, dimension( \* ) work, double precision, dimension( \* ) rwork, integer info) **ZHPRFS** **Purpose:** ``` !> !> ZHPRFS improves the computed solution to a system of linear !> equations when the coefficient matrix is Hermitian indefinite !> and packed, and provides error bounds and backward error estimates !> for the solution. !> ``` **Parameters** *UPLO* ``` !> UPLO is CHARACTER*1 !> = 'U': Upper triangle of A is stored; !> = 'L': Lower triangle of A is stored. !> ``` *N* ``` !> N is INTEGER !> The order of the matrix A. N >= 0. !> ``` *NRHS* ``` !> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrices B and X. NRHS >= 0. !> ``` *AP* ``` !> AP is COMPLEX*16 array, dimension (N*(N+1)/2) !> The upper or lower triangle of the Hermitian 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)*(2*n-j)/2) = A(i,j) for j<=i<=n. !> ``` *AFP* ``` !> AFP is COMPLEX*16 array, dimension (N*(N+1)/2) !> The factored form of the matrix A. AFP contains the block !> diagonal matrix D and the multipliers used to obtain the !> factor U or L from the factorization A = U*D*U**H or !> A = L*D*L**H as computed by ZHPTRF, stored as a packed !> triangular matrix. !> ``` *IPIV* ``` !> IPIV is INTEGER array, dimension (N) !> Details of the interchanges and the block structure of D !> as determined by ZHPTRF. !> ``` *B* ``` !> B is COMPLEX*16 array, dimension (LDB,NRHS) !> The right hand side matrix B. !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,N). !> ``` *X* ``` !> X is COMPLEX*16 array, dimension (LDX,NRHS) !> On entry, the solution matrix X, as computed by ZHPTRS. !> On exit, the improved solution matrix X. !> ``` *LDX* ``` !> LDX is INTEGER !> The leading dimension of the array X. LDX >= max(1,N). !> ``` *FERR* ``` !> FERR is DOUBLE PRECISION array, dimension (NRHS) !> The estimated forward error bound 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) !> is an estimated upper bound for the magnitude of the largest !> element in (X(j) - XTRUE) divided by the magnitude of the !> largest element in X(j). The estimate is as reliable as !> the estimate for RCOND, and is almost always a slight !> overestimate of the true error. !> ``` *BERR* ``` !> BERR is 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* ``` !> WORK is COMPLEX*16 array, dimension (2*N) !> ``` *RWORK* ``` !> RWORK is DOUBLE PRECISION array, dimension (N) !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> ``` **Internal Parameters:** ``` !> ITMAX is the maximum number of steps of iterative refinement. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **178** of file **zhprfs.f**. ### subroutine zsprfs (character uplo, integer n, integer nrhs, complex\*16, dimension( \* ) ap, complex\*16, dimension( \* ) afp, integer, dimension( \* ) ipiv, complex\*16, dimension( ldb, \* ) b, integer ldb, complex\*16, dimension( ldx, \* ) x, integer ldx, double precision, dimension( \* ) ferr, double precision, dimension( \* ) berr, complex\*16, dimension( \* ) work, double precision, dimension( \* ) rwork, integer info) **ZSPRFS** **Purpose:** ``` !> !> ZSPRFS improves the computed solution to a system of linear !> equations when the coefficient matrix is symmetric indefinite !> and packed, and provides error bounds and backward error estimates !> for the solution. !> ``` **Parameters** *UPLO* ``` !> UPLO is CHARACTER*1 !> = 'U': Upper triangle of A is stored; !> = 'L': Lower triangle of A is stored. !> ``` *N* ``` !> N is INTEGER !> The order of the matrix A. N >= 0. !> ``` *NRHS* ``` !> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrices B and X. NRHS >= 0. !> ``` *AP* ``` !> AP is COMPLEX*16 array, dimension (N*(N+1)/2) !> The upper or lower triangle of the symmetric 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)*(2*n-j)/2) = A(i,j) for j<=i<=n. !> ``` *AFP* ``` !> AFP is COMPLEX*16 array, dimension (N*(N+1)/2) !> The factored form of the matrix A. AFP contains the block !> diagonal matrix D and the multipliers used to obtain the !> factor U or L from the factorization A = U*D*U**T or !> A = L*D*L**T as computed by ZSPTRF, stored as a packed !> triangular matrix. !> ``` *IPIV* ``` !> IPIV is INTEGER array, dimension (N) !> Details of the interchanges and the block structure of D !> as determined by ZSPTRF. !> ``` *B* ``` !> B is COMPLEX*16 array, dimension (LDB,NRHS) !> The right hand side matrix B. !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,N). !> ``` *X* ``` !> X is COMPLEX*16 array, dimension (LDX,NRHS) !> On entry, the solution matrix X, as computed by ZSPTRS. !> On exit, the improved solution matrix X. !> ``` *LDX* ``` !> LDX is INTEGER !> The leading dimension of the array X. LDX >= max(1,N). !> ``` *FERR* ``` !> FERR is DOUBLE PRECISION array, dimension (NRHS) !> The estimated forward error bound 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) !> is an estimated upper bound for the magnitude of the largest !> element in (X(j) - XTRUE) divided by the magnitude of the !> largest element in X(j). The estimate is as reliable as !> the estimate for RCOND, and is almost always a slight !> overestimate of the true error. !> ``` *BERR* ``` !> BERR is 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* ``` !> WORK is COMPLEX*16 array, dimension (2*N) !> ``` *RWORK* ``` !> RWORK is DOUBLE PRECISION array, dimension (N) !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> ``` **Internal Parameters:** ``` !> ITMAX is the maximum number of steps of iterative refinement. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **178** of file **zsprfs.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.