--- name: "hptrd" 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/hptrd" --- # hptrd(3) {hp,sp}trd: reduction to tridiagonal ## NAME hptrd - {hp,sp}trd: reduction to tridiagonal ## SYNOPSIS ### Functions subroutine **chptrd** (uplo, n, ap, d, e, tau, info)\ **CHPTRD** subroutine **dsptrd** (uplo, n, ap, d, e, tau, info)\ **DSPTRD** subroutine **ssptrd** (uplo, n, ap, d, e, tau, info)\ **SSPTRD** subroutine **zhptrd** (uplo, n, ap, d, e, tau, info)\ **ZHPTRD** ## Detailed Description ## Function Documentation ### subroutine chptrd (character uplo, integer n, complex, dimension( \* ) ap, real, dimension( \* ) d, real, dimension( \* ) e, complex, dimension( \* ) tau, integer info) **CHPTRD** **Purpose:** ``` !> !> CHPTRD reduces a complex Hermitian matrix A stored in packed form to !> real symmetric tridiagonal form T by a unitary similarity !> transformation: Q**H * A * Q = T. !> ``` **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. !> ``` *AP* ``` !> AP is COMPLEX array, dimension (N*(N+1)/2) !> On entry, 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. !> On exit, if UPLO = 'U', the diagonal and first superdiagonal !> of A are overwritten by the corresponding elements of the !> tridiagonal matrix T, and the elements above the first !> superdiagonal, with the array TAU, represent the unitary !> matrix Q as a product of elementary reflectors; if UPLO !> = 'L', the diagonal and first subdiagonal of A are over- !> written by the corresponding elements of the tridiagonal !> matrix T, and the elements below the first subdiagonal, with !> the array TAU, represent the unitary matrix Q as a product !> of elementary reflectors. See Further Details. !> ``` *D* ``` !> D is REAL array, dimension (N) !> The diagonal elements of the tridiagonal matrix T: !> D(i) = A(i,i). !> ``` *E* ``` !> E is REAL array, dimension (N-1) !> The off-diagonal elements of the tridiagonal matrix T: !> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. !> ``` *TAU* ``` !> TAU is COMPLEX array, dimension (N-1) !> The scalar factors of the elementary reflectors (see Further !> Details). !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. **Further Details:** ``` !> !> If UPLO = 'U', the matrix Q is represented as a product of elementary !> reflectors !> !> Q = H(n-1) . . . H(2) H(1). !> !> Each H(i) has the form !> !> H(i) = I - tau * v * v**H !> !> where tau is a complex scalar, and v is a complex vector with !> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP, !> overwriting A(1:i-1,i+1), and tau is stored in TAU(i). !> !> If UPLO = 'L', the matrix Q is represented as a product of elementary !> reflectors !> !> Q = H(1) H(2) . . . H(n-1). !> !> Each H(i) has the form !> !> H(i) = I - tau * v * v**H !> !> where tau is a complex scalar, and v is a complex vector with !> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP, !> overwriting A(i+2:n,i), and tau is stored in TAU(i). !> ``` Definition at line **150** of file **chptrd.f**. ### subroutine dsptrd (character uplo, integer n, double precision, dimension( \* ) ap, double precision, dimension( \* ) d, double precision, dimension( \* ) e, double precision, dimension( \* ) tau, integer info) **DSPTRD** **Purpose:** ``` !> !> DSPTRD reduces a real symmetric matrix A stored in packed form to !> symmetric tridiagonal form T by an orthogonal similarity !> transformation: Q**T * A * Q = T. !> ``` **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. !> ``` *AP* ``` !> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2) !> On entry, 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. !> On exit, if UPLO = 'U', the diagonal and first superdiagonal !> of A are overwritten by the corresponding elements of the !> tridiagonal matrix T, and the elements above the first !> superdiagonal, with the array TAU, represent the orthogonal !> matrix Q as a product of elementary reflectors; if UPLO !> = 'L', the diagonal and first subdiagonal of A are over- !> written by the corresponding elements of the tridiagonal !> matrix T, and the elements below the first subdiagonal, with !> the array TAU, represent the orthogonal matrix Q as a product !> of elementary reflectors. See Further Details. !> ``` *D* ``` !> D is DOUBLE PRECISION array, dimension (N) !> The diagonal elements of the tridiagonal matrix T: !> D(i) = A(i,i). !> ``` *E* ``` !> E is DOUBLE PRECISION array, dimension (N-1) !> The off-diagonal elements of the tridiagonal matrix T: !> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. !> ``` *TAU* ``` !> TAU is DOUBLE PRECISION array, dimension (N-1) !> The scalar factors of the elementary reflectors (see Further !> Details). !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. **Further Details:** ``` !> !> If UPLO = 'U', the matrix Q is represented as a product of elementary !> reflectors !> !> Q = H(n-1) . . . H(2) H(1). !> !> Each H(i) has the form !> !> H(i) = I - tau * v * v**T !> !> where tau is a real scalar, and v is a real vector with !> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP, !> overwriting A(1:i-1,i+1), and tau is stored in TAU(i). !> !> If UPLO = 'L', the matrix Q is represented as a product of elementary !> reflectors !> !> Q = H(1) H(2) . . . H(n-1). !> !> Each H(i) has the form !> !> H(i) = I - tau * v * v**T !> !> where tau is a real scalar, and v is a real vector with !> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP, !> overwriting A(i+2:n,i), and tau is stored in TAU(i). !> ``` Definition at line **149** of file **dsptrd.f**. ### subroutine ssptrd (character uplo, integer n, real, dimension( \* ) ap, real, dimension( \* ) d, real, dimension( \* ) e, real, dimension( \* ) tau, integer info) **SSPTRD** **Purpose:** ``` !> !> SSPTRD reduces a real symmetric matrix A stored in packed form to !> symmetric tridiagonal form T by an orthogonal similarity !> transformation: Q**T * A * Q = T. !> ``` **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. !> ``` *AP* ``` !> AP is REAL array, dimension (N*(N+1)/2) !> On entry, 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. !> On exit, if UPLO = 'U', the diagonal and first superdiagonal !> of A are overwritten by the corresponding elements of the !> tridiagonal matrix T, and the elements above the first !> superdiagonal, with the array TAU, represent the orthogonal !> matrix Q as a product of elementary reflectors; if UPLO !> = 'L', the diagonal and first subdiagonal of A are over- !> written by the corresponding elements of the tridiagonal !> matrix T, and the elements below the first subdiagonal, with !> the array TAU, represent the orthogonal matrix Q as a product !> of elementary reflectors. See Further Details. !> ``` *D* ``` !> D is REAL array, dimension (N) !> The diagonal elements of the tridiagonal matrix T: !> D(i) = A(i,i). !> ``` *E* ``` !> E is REAL array, dimension (N-1) !> The off-diagonal elements of the tridiagonal matrix T: !> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. !> ``` *TAU* ``` !> TAU is REAL array, dimension (N-1) !> The scalar factors of the elementary reflectors (see Further !> Details). !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. **Further Details:** ``` !> !> If UPLO = 'U', the matrix Q is represented as a product of elementary !> reflectors !> !> Q = H(n-1) . . . H(2) H(1). !> !> Each H(i) has the form !> !> H(i) = I - tau * v * v**T !> !> where tau is a real scalar, and v is a real vector with !> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP, !> overwriting A(1:i-1,i+1), and tau is stored in TAU(i). !> !> If UPLO = 'L', the matrix Q is represented as a product of elementary !> reflectors !> !> Q = H(1) H(2) . . . H(n-1). !> !> Each H(i) has the form !> !> H(i) = I - tau * v * v**T !> !> where tau is a real scalar, and v is a real vector with !> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP, !> overwriting A(i+2:n,i), and tau is stored in TAU(i). !> ``` Definition at line **149** of file **ssptrd.f**. ### subroutine zhptrd (character uplo, integer n, complex\*16, dimension( \* ) ap, double precision, dimension( \* ) d, double precision, dimension( \* ) e, complex\*16, dimension( \* ) tau, integer info) **ZHPTRD** **Purpose:** ``` !> !> ZHPTRD reduces a complex Hermitian matrix A stored in packed form to !> real symmetric tridiagonal form T by a unitary similarity !> transformation: Q**H * A * Q = T. !> ``` **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. !> ``` *AP* ``` !> AP is COMPLEX*16 array, dimension (N*(N+1)/2) !> On entry, 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. !> On exit, if UPLO = 'U', the diagonal and first superdiagonal !> of A are overwritten by the corresponding elements of the !> tridiagonal matrix T, and the elements above the first !> superdiagonal, with the array TAU, represent the unitary !> matrix Q as a product of elementary reflectors; if UPLO !> = 'L', the diagonal and first subdiagonal of A are over- !> written by the corresponding elements of the tridiagonal !> matrix T, and the elements below the first subdiagonal, with !> the array TAU, represent the unitary matrix Q as a product !> of elementary reflectors. See Further Details. !> ``` *D* ``` !> D is DOUBLE PRECISION array, dimension (N) !> The diagonal elements of the tridiagonal matrix T: !> D(i) = A(i,i). !> ``` *E* ``` !> E is DOUBLE PRECISION array, dimension (N-1) !> The off-diagonal elements of the tridiagonal matrix T: !> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. !> ``` *TAU* ``` !> TAU is COMPLEX*16 array, dimension (N-1) !> The scalar factors of the elementary reflectors (see Further !> Details). !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. **Further Details:** ``` !> !> If UPLO = 'U', the matrix Q is represented as a product of elementary !> reflectors !> !> Q = H(n-1) . . . H(2) H(1). !> !> Each H(i) has the form !> !> H(i) = I - tau * v * v**H !> !> where tau is a complex scalar, and v is a complex vector with !> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP, !> overwriting A(1:i-1,i+1), and tau is stored in TAU(i). !> !> If UPLO = 'L', the matrix Q is represented as a product of elementary !> reflectors !> !> Q = H(1) H(2) . . . H(n-1). !> !> Each H(i) has the form !> !> H(i) = I - tau * v * v**H !> !> where tau is a complex scalar, and v is a complex vector with !> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP, !> overwriting A(i+2:n,i), and tau is stored in TAU(i). !> ``` Definition at line **150** of file **zhptrd.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.