hbtrd - {hb,sb}trd: reduction to tridiagonal

Generated automatically by Doxygen for LAPACK from the source code. Version 3.12.0 Thu Aug 7 2025 17:26:25 hbtrd(3)

subroutinechbtrd(charactervect,characteruplo,integern,integerkd,complex,dimension(ldab,*)ab,integerldab,real,dimension(*)d,real,dimension(*)e,complex,dimension(ldq,*)q,integerldq,complex,dimension(*)work,integerinfo)CHBTRDPurpose: CHBTRD reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T. ParametersVECT VECT is CHARACTER*1 = 'N': do not form Q; = 'V': form Q; = 'U': update a matrix X, by forming X*Q. 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. KD KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0. AB AB is COMPLEX array, dimension (LDAB,N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, the diagonal elements of AB are overwritten by the diagonal elements of the tridiagonal matrix T; if KD > 0, the elements on the first superdiagonal (if UPLO = 'U') or the first subdiagonal (if UPLO = 'L') are overwritten by the off-diagonal elements of T; the rest of AB is overwritten by values generated during the reduction. LDAB LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1. D D is REAL array, dimension (N) The diagonal elements of the tridiagonal matrix T. E E is REAL array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T: E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'. Q Q is COMPLEX array, dimension (LDQ,N) On entry, if VECT = 'U', then Q must contain an N-by-N matrix X; if VECT = 'N' or 'V', then Q need not be set. On exit: if VECT = 'V', Q contains the N-by-N unitary matrix Q; if VECT = 'U', Q contains the product X*Q; if VECT = 'N', the array Q is not referenced. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'. WORK WORK is COMPLEX array, dimension (N) 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. FurtherDetails: Modified by Linda Kaufman, Bell Labs. subroutinedsbtrd(charactervect,characteruplo,integern,integerkd,doubleprecision,dimension(ldab,*)ab,integerldab,doubleprecision,dimension(*)d,doubleprecision,dimension(*)e,doubleprecision,dimension(ldq,*)q,integerldq,doubleprecision,dimension(*)work,integerinfo)DSBTRDPurpose: DSBTRD reduces a real symmetric band matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T. ParametersVECT VECT is CHARACTER*1 = 'N': do not form Q; = 'V': form Q; = 'U': update a matrix X, by forming X*Q. 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. KD KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0. AB AB is DOUBLE PRECISION array, dimension (LDAB,N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, the diagonal elements of AB are overwritten by the diagonal elements of the tridiagonal matrix T; if KD > 0, the elements on the first superdiagonal (if UPLO = 'U') or the first subdiagonal (if UPLO = 'L') are overwritten by the off-diagonal elements of T; the rest of AB is overwritten by values generated during the reduction. LDAB LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1. D D is DOUBLE PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T. E E is DOUBLE PRECISION array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T: E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'. Q Q is DOUBLE PRECISION array, dimension (LDQ,N) On entry, if VECT = 'U', then Q must contain an N-by-N matrix X; if VECT = 'N' or 'V', then Q need not be set. On exit: if VECT = 'V', Q contains the N-by-N orthogonal matrix Q; if VECT = 'U', Q contains the product X*Q; if VECT = 'N', the array Q is not referenced. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'. WORK WORK 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 Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. FurtherDetails: Modified by Linda Kaufman, Bell Labs. subroutinessbtrd(charactervect,characteruplo,integern,integerkd,real,dimension(ldab,*)ab,integerldab,real,dimension(*)d,real,dimension(*)e,real,dimension(ldq,*)q,integerldq,real,dimension(*)work,integerinfo)SSBTRDPurpose: SSBTRD reduces a real symmetric band matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T. ParametersVECT VECT is CHARACTER*1 = 'N': do not form Q; = 'V': form Q; = 'U': update a matrix X, by forming X*Q. 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. KD KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0. AB AB is REAL array, dimension (LDAB,N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, the diagonal elements of AB are overwritten by the diagonal elements of the tridiagonal matrix T; if KD > 0, the elements on the first superdiagonal (if UPLO = 'U') or the first subdiagonal (if UPLO = 'L') are overwritten by the off-diagonal elements of T; the rest of AB is overwritten by values generated during the reduction. LDAB LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1. D D is REAL array, dimension (N) The diagonal elements of the tridiagonal matrix T. E E is REAL array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T: E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'. Q Q is REAL array, dimension (LDQ,N) On entry, if VECT = 'U', then Q must contain an N-by-N matrix X; if VECT = 'N' or 'V', then Q need not be set. On exit: if VECT = 'V', Q contains the N-by-N orthogonal matrix Q; if VECT = 'U', Q contains the product X*Q; if VECT = 'N', the array Q is not referenced. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'. WORK WORK is REAL array, dimension (N) 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. FurtherDetails: Modified by Linda Kaufman, Bell Labs. subroutinezhbtrd(charactervect,characteruplo,integern,integerkd,complex*16,dimension(ldab,*)ab,integerldab,doubleprecision,dimension(*)d,doubleprecision,dimension(*)e,complex*16,dimension(ldq,*)q,integerldq,complex*16,dimension(*)work,integerinfo)ZHBTRDPurpose: ZHBTRD reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T. ParametersVECT VECT is CHARACTER*1 = 'N': do not form Q; = 'V': form Q; = 'U': update a matrix X, by forming X*Q. 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. KD KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0. AB AB is COMPLEX*16 array, dimension (LDAB,N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, the diagonal elements of AB are overwritten by the diagonal elements of the tridiagonal matrix T; if KD > 0, the elements on the first superdiagonal (if UPLO = 'U') or the first subdiagonal (if UPLO = 'L') are overwritten by the off-diagonal elements of T; the rest of AB is overwritten by values generated during the reduction. LDAB LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1. D D is DOUBLE PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T. E E is DOUBLE PRECISION array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T: E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'. Q Q is COMPLEX*16 array, dimension (LDQ,N) On entry, if VECT = 'U', then Q must contain an N-by-N matrix X; if VECT = 'N' or 'V', then Q need not be set. On exit: if VECT = 'V', Q contains the N-by-N unitary matrix Q; if VECT = 'U', Q contains the product X*Q; if VECT = 'N', the array Q is not referenced. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'. WORK WORK is COMPLEX*16 array, dimension (N) 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. FurtherDetails: Modified by Linda Kaufman, Bell Labs.

hbtrd - {hb,sb}trd: reduction to tridiagonal

Functions subroutine chbtrd (vect, uplo, n, kd, ab, ldab, d, e, q, ldq, work, info) CHBTRD subroutine dsbtrd (vect, uplo, n, kd, ab, ldab, d, e, q, ldq, work, info) DSBTRD subroutine ssbtrd (vect, uplo, n, kd, ab, ldab, d, e, q, ldq, work, info) SSBTRD subroutine zhbtrd (vect, uplo, n, kd, ab, ldab, d, e, q, ldq, work, info) ZHBTRD