Module Slap.D

module D: Slap_D
64-bit real BLAS and LAPACK functions.

type prec = Bigarray.float64_elt 
type num_type = float 
type ('indim, 'outdim, [< `N | `T ]) trans3 = ('indim, 'outdim, [< `N | `T ] as 'a) Slap_common.trans2 
A type of transpose parameters (Slap_common.normal and Slap_common.trans). For complex matrices, Slap_common.conjtr is also offered, hence the name.
type ('n, 'cnt_or_dsc) vec = ('n, num_type, prec, 'cnt_or_dsc) Slap_vec.t 
Vectors.
type ('m, 'n, 'cnt_or_dsc) mat = ('m, 'n, num_type, prec, 'cnt_or_dsc) Slap_mat.t 
Matrices.
type rprec = Bigarray.float64_elt 
type ('n, 'cnt_or_dsc) rvec = ('n, float, rprec, 'cnt_or_dsc) Slap_vec.t 
Real vectors. (In Slap.S and Slap.D, rvec is equal to vec.)
val prec : (num_type, prec) Bigarray.kind
val rprec : (float, rprec) Bigarray.kind
module Vec: sig .. end
module Mat: sig .. end
val pp_num : Format.formatter -> num_type -> unit
A pretty-printer for elements in vectors and matrices.
val pp_vec : Format.formatter -> ('n, 'cnt_or_dsc) vec -> unit
A pretty-printer for column vectors.
val pp_mat : Format.formatter -> ('m, 'n, 'cnt_or_dsc) mat -> unit
A pretty-printer for matrices.

BLAS interface



BLAS interface



Level 1


val swap : ('n, 'x_cd) vec -> ('n, 'y_cd) vec -> unit
swap x y swaps elements in x and y.
val scal : num_type -> ('n, 'cd) vec -> unit
scal c x multiplies all elements in x by scalar value c, and destructively assigns the result to x.
val copy : ?y:('n, 'y_cd) vec -> ('n, 'x_cd) vec -> ('n, 'y_cd) vec
copy ?y x copies x into y.
Returns vector y, which is overwritten.
val nrm2 : ('n, 'cd) vec -> float
nrm2 x retruns the L2 norm of vector x: ||x||.
val axpy : ?alpha:num_type ->
('n, 'x_cd) vec -> ('n, 'y_cd) vec -> unit
axpy ?alpha x y executes y := alpha * x + y with scalar value alpha, and vectors x and y.
alpha : default = 1.0
val iamax : ('n, 'cd) vec -> int
iamax x returns the index of the maximum value of all elements in x.
val amax : ('n, 'cd) vec -> num_type
amax x finds the maximum value of all elements in x.

Level 2


val gemv : ?beta:num_type ->
?y:('m, 'y_cd) vec ->
trans:('a_m * 'a_n, 'm * 'n, [< `N | `T ]) trans3 ->
?alpha:num_type ->
('a_m, 'a_n, 'a_cd) mat ->
('n, 'x_cd) vec -> ('m, 'y_cd) vec
gemv ?beta ?y ~trans ?alpha a x executes y := alpha * OP(a) * x + beta * y.
beta : default = 0.0
trans : the transpose flag for a:
alpha : default = 1.0
val gbmv : m:'a_m Slap_size.t ->
?beta:num_type ->
?y:('m, 'y_cd) vec ->
trans:('a_m * 'a_n, 'm * 'n, [< `N | `T ]) trans3 ->
?alpha:num_type ->
(('a_m, 'a_n, 'kl, 'ku) Slap_size.geband, 'a_n, 'a_cd) mat ->
'kl Slap_size.t ->
'ku Slap_size.t -> ('n, 'x_cd) vec -> ('m, 'y_cd) vec
gbmv ~m ?beta ?y ~trans ?alpha a kl ku x computes y := alpha * OP(a) * x + beta * y where a is a band matrix stored in band storage.
Since 0.2.0
Returns vector y, which is overwritten.
beta : default = 0.0
trans : the transpose flag for a:
alpha : default = 1.0
val symv : ?beta:num_type ->
?y:('n, 'y_cd) vec ->
?up:[< `L | `U ] Slap_common.uplo ->
?alpha:num_type ->
('n, 'n, 'a_cd) mat ->
('n, 'x_cd) vec -> ('n, 'y_cd) vec
symv ?beta ?y ?up ?alpha a x executes y := alpha * a * x + beta * y.
beta : default = 0.0
up : default = Slap_common.upper
alpha : default = 1.0
val trmv : trans:('n * 'n, 'n * 'n, [< `N | `T ]) trans3 ->
?diag:Slap_common.diag ->
?up:[< `L | `U ] Slap_common.uplo ->
('n, 'n, 'a_cd) mat -> ('n, 'x_cd) vec -> unit
trmv ~trans ?diag ?up a x executes x := OP(a) * x.
trans : the transpose flag for a:
diag : default = Slap_common.non_unit
up : default = Slap_common.upper
val trsv : trans:('n * 'n, 'n * 'n, [< `N | `T ]) trans3 ->
?diag:Slap_common.diag ->
?up:[< `L | `U ] Slap_common.uplo ->
('n, 'n, 'a_cd) mat -> ('n, 'b_cd) vec -> unit
trmv ~trans ?diag ?up a b solves linear system OP(a) * x = b and destructively assigns x to b.
trans : the transpose flag for a:
diag : default = Slap_common.non_unit
up : default = Slap_common.upper
val tpmv : trans:('n * 'n, 'n * 'n, [< `N | `T ]) trans3 ->
?diag:Slap_common.diag ->
?up:[< `L | `U ] Slap_common.uplo ->
('n Slap_size.packed, Slap_misc.cnt) vec ->
('n, 'x_cd) vec -> unit
tpmv ~trans ?diag ?up a x executes x := OP(a) * x where a is a packed triangular matrix.
Since 0.2.0
trans : the transpose flag for a:
diag : default = Slap_common.non_unit
up : default = Slap_common.upper
val tpsv : trans:('n * 'n, 'n * 'n, [< `N | `T ]) trans3 ->
?diag:Slap_common.diag ->
?up:[< `L | `U ] Slap_common.uplo ->
('n Slap_size.packed, Slap_misc.cnt) vec ->
('n, 'x_cd) vec -> unit
tpsv ~trans ?diag ?up a b solves linear system OP(a) * x = b and destructively assigns x to b where a is a packed triangular matrix.
Since 0.2.0
trans : the transpose flag for a:
diag : default = Slap_common.non_unit
up : default = Slap_common.upper

Level 3


val gemm : ?beta:num_type ->
?c:('m, 'n, 'c_cd) mat ->
transa:('a_m * 'a_k, 'm * 'k, [< `N | `T ]) trans3 ->
?alpha:num_type ->
('a_m, 'a_k, 'a_cd) mat ->
transb:('b_k * 'b_n, 'k * 'n, [< `N | `T ]) trans3 ->
('b_k, 'b_n, 'b_cd) mat -> ('m, 'n, 'c_cd) mat
gemm ?beta ?c ~transa ?alpha a ~transb b executes c := alpha * OP(a) * OP(b) + beta * c.
beta : default = 0.0
transa : the transpose flag for a:
alpha : default = 1.0
transb : the transpose flag for b:
val symm : side:('k, 'm, 'n) Slap_common.side ->
?up:[< `L | `U ] Slap_common.uplo ->
?beta:num_type ->
?c:('m, 'n, 'c_cd) mat ->
?alpha:num_type ->
('k, 'k, 'a_cd) mat ->
('m, 'n, 'b_cd) mat -> ('m, 'n, 'c_cd) mat
symm ~side ?up ?beta ?c ?alpha a b executes

where a is a symmterix matrix, and b and c are general matrices.
side : the side flag to specify direction of multiplication of a and b.
up : default = Slap_common.upper
beta : default = 0.0
alpha : default = 1.0
val trmm : side:('k, 'm, 'n) Slap_common.side ->
?up:[< `L | `U ] Slap_common.uplo ->
transa:('k * 'k, 'k * 'k, [< `N | `T ]) trans3 ->
?diag:Slap_common.diag ->
?alpha:num_type ->
a:('k, 'k, 'a_cd) mat -> ('m, 'n, 'b_cd) mat -> unit
trmm ~side ?up ~transa ?diag ?alpha ~a b executes

where a is a triangular matrix, and b is a general matrix.
side : the side flag to specify direction of multiplication of a and b.
up : default = Slap_common.upper
transa : the transpose flag for a:
diag : default = Slap_common.non_unit
alpha : default = 1.0
val trsm : side:('k, 'm, 'n) Slap_common.side ->
?up:[< `L | `U ] Slap_common.uplo ->
transa:('k * 'k, 'k * 'k, [< `N | `T ]) trans3 ->
?diag:Slap_common.diag ->
?alpha:num_type ->
a:('k, 'k, 'a_cd) mat -> ('m, 'n, 'b_cd) mat -> unit
trsm ~side ?up ~transa ?diag ?alpha ~a b solves a system of linear equations

where a is a triangular matrix, and b is a general matrix. The solution x is returned by b.
side : the side flag to specify direction of multiplication of a and b.
up : default = Slap_common.upper
transa : the transpose flag for a:
diag : default = Slap_common.non_unit
alpha : default = 1.0
val syrk : ?up:[< `L | `U ] Slap_common.uplo ->
?beta:num_type ->
?c:('n, 'n, 'c_cd) mat ->
trans:('a_n * 'a_k, 'n * 'k, [< `N | `T ]) Slap_common.trans2 ->
?alpha:num_type ->
('a_n, 'a_k, 'a_cd) mat -> ('n, 'n, 'c_cd) mat
syrk ?up ?beta ?c ~trans ?alpha a executes

where a is a general matrix and c is a symmetric matrix.
up : default = Slap_common.upper
beta : default = 0.0
trans : the transpose flag for a
alpha : default = 1.0
val syr2k : ?up:[< `L | `U ] Slap_common.uplo ->
?beta:num_type ->
?c:('n, 'n, 'c_cd) mat ->
trans:('p * 'q, 'n * 'k, [< `N | `T ]) Slap_common.trans2 ->
?alpha:num_type ->
('p, 'q, 'a_cd) mat ->
('p, 'q, 'b_cd) mat -> ('n, 'n, 'c_cd) mat
syr2k ?up ?beta ?c ~trans ?alpha a b computes

with symmetric matrix c, and general matrices a and b.
up : default = Slap_common.upper
beta : default = 0.0
trans : the transpose flag for a
alpha : default = 1.0

LAPACK interface



LAPACK interface



Auxiliary routines


val lacpy : ?uplo:[< `A | `L | `U ] Slap_common.uplo ->
?b:('m, 'n, 'b_cd) mat ->
('m, 'n, 'a_cd) mat -> ('m, 'n, 'b_cd) mat
lacpy ?uplo ?b a copies the matrix a into the matrix b.
Returns b, which is overwritten.
uplo : default = Slap_common.upper_lower
b : default = a fresh matrix.
val lassq : ?scale:float -> ?sumsq:float -> ('n, 'cd) vec -> float * float
lassq ?scale ?sumsq x
Returns (scl, smsq) where scl and smsq satisfy scl^2 * smsq = x1^2 + x2^2 + ... + xn^2 + scale^2 * smsq.
scale : default = 0.0
sumsq : default = 1.0
type larnv_liseed = Slap_size.four 
val larnv : ?idist:[ `Normal | `Uniform0 | `Uniform1 ] ->
?iseed:(larnv_liseed, Slap_misc.cnt) Slap_common.int32_vec ->
x:('n, Slap_misc.cnt) vec -> unit -> ('n, 'cnt) vec
larnv ?idist ?iseed ~x () generates a random vector with the random distribution specified by idist and random seed iseed.
Returns vector x, which is overwritten.
idist : default = `Normal
iseed : a four-dimensional integer vector with all ones.
type ('m, 'a) lange_min_lwork 
val lange_min_lwork : 'm Slap_size.t ->
([< `F | `I | `M | `O ] as 'a) Slap_common.norm4 ->
('m, 'a) lange_min_lwork Slap_size.t
lange_min_lwork m norm computes the minimum length of workspace for lange routine. m is the number of rows in a matrix, and norm is the sort of matrix norms.
val lange : ?norm:[< `F | `I | `M | `O ] Slap_common.norm4 ->
?work:('lwork, Slap_misc.cnt) rvec ->
('m, 'n, 'cd) mat -> float
lange ?norm ?work a
Returns the norm of matrix a.
norm : default = Slap_common.norm_1.
work : default = an optimum-length vector.
val lauum : ?up:[< `L | `U ] Slap_common.uplo -> ('n, 'n, 'cd) mat -> unit
lauum ?up a computes

The upper or lower triangular part is overwritten.
up : default = Slap_common.upper

Linear equations (computational routines)


val getrf : ?ipiv:(('m, 'n) Slap_size.min, Slap_misc.cnt) Slap_common.int32_vec ->
('m, 'n, 'cd) mat ->
(('m, 'n) Slap_size.min, 'cnt) Slap_common.int32_vec
getrf ?ipiv a computes LU factorization of matrix a using partial pivoting with row interchanges: a = P * L * U where P is a permutation matrix, and L and U are lower and upper triangular matrices, respectively. the permutation matrix is returned in ipiv.
Raises Failure if the matrix is singular.
Returns vector ipiv, which is overwritten.
val getrs : ?ipiv:(('n, 'n) Slap_size.min, Slap_misc.cnt) Slap_common.int32_vec ->
trans:('n * 'n, 'n * 'n, [< `N | `T ]) trans3 ->
('n, 'n, 'a_cd) mat -> ('n, 'n, 'b_cd) mat -> unit
getrs ?ipiv trans a b solves systems of linear equations OP(a) * x = b where a a 'n-by-'n general matrix, each column of matrix b is the r.h.s. vector, and each column of matrix x is the corresponding solution. The solution x is returned in b.
Raises Failure if the matrix is singular.
ipiv : a result of gesv or getrf. It is internally computed by getrf if omitted.
trans : the transpose flag for a:
type 'n getri_min_lwork 
val getri_min_lwork : 'n Slap_size.t -> 'n getri_min_lwork Slap_size.t
getri_min_lwork n computes the minimum length of workspace for getri routine. n is the number of columns or rows in a matrix.
val getri_opt_lwork : ('n, 'n, 'cd) mat -> (module Slap_size.SIZE)
getri_opt_lwork a computes the optimal length of workspace for getri routine.
val getri : ?ipiv:(('n, 'n) Slap_size.min, Slap_misc.cnt) Slap_common.int32_vec ->
?work:('lwork, Slap_misc.cnt) vec -> ('n, 'n, 'cd) mat -> unit
getri ?ipiv ?work a computes the inverse of general matrix a by LU-factorization. The inverse matrix is returned in a.
Raises Failure if the matrix is singular.
ipiv : a result of gesv or getrf. It is internally computed by getrf if omitted.
work : default = an optimum-length vector.
type sytrf_min_lwork 
val sytrf_min_lwork : unit -> sytrf_min_lwork Slap_size.t
sytrf_min_lwork () computes the minimum length of workspace for sytrf routine.
val sytrf_opt_lwork : ?up:[< `L | `U ] Slap_common.uplo ->
('n, 'n, 'cd) mat -> (module Slap_size.SIZE)
sytrf_opt_lwork ?up a computes the optimal length of workspace for sytrf routine.
up : default = true
val sytrf : ?up:[< `L | `U ] Slap_common.uplo ->
?ipiv:('n, Slap_misc.cnt) Slap_common.int32_vec ->
?work:('lwork, Slap_misc.cnt) vec ->
('n, 'n, 'cd) mat -> ('n, 'cnt) Slap_common.int32_vec
sytrf ?up ?ipiv ?work a factorizes symmetric matrix a using the Bunch-Kaufman diagonal pivoting method:

where P is a permutation matrix, U and L are upper and lower triangular matrices with unit diagonal, and D is a symmetric block-diagonal matrix. The permutation matrix is returned in ipiv.
Raises Failure if a is singular.
Returns vector ipiv, which is overwritten.
up : default = Slap_common.upper
work : default = an optimum-length vector.
val sytrs : ?up:[< `L | `U ] Slap_common.uplo ->
?ipiv:('n, Slap_misc.cnt) Slap_common.int32_vec ->
('n, 'n, 'a_cd) mat -> ('n, 'nrhs, 'b_cd) mat -> unit
sytrs ?up ?ipiv a b solves systems of linear equations a * x = b where a is a symmetric matrix, each column of matrix b is the r.h.s. vector, and each column of matrix x is the corresponding solution. The solution x is returned in b.

This routine uses the Bunch-Kaufman diagonal pivoting method:

where P is a permutation matrix, U and L are upper and lower triangular matrices with unit diagonal, and D is a symmetric block-diagonal matrix.
Raises Failure if a is singular.
up : default = Slap_common.upper
ipiv : a result of sytrf. It is internally computed by sytrf if omitted.
type 'n sytri_min_lwork 
val sytri_min_lwork : 'n Slap_size.t -> 'n sytri_min_lwork Slap_size.t
sytri_min_lwork () computes the minimum length of workspace for sytri routine.
val sytri : ?up:[< `L | `U ] Slap_common.uplo ->
?ipiv:('n, Slap_misc.cnt) Slap_common.int32_vec ->
?work:('lwork, Slap_misc.cnt) vec -> ('n, 'n, 'cd) mat -> unit
sytri ?up ?ipiv ?work a computes the inverse of symmetric matrix a using the Bunch-Kaufman diagonal pivoting method:

where P is a permutation matrix, U and L are upper and lower triangular matrices with unit diagonal, and D is a symmetric block-diagonal matrix.
Raises Failure if a is singular.
up : default = Slap_common.upper
ipiv : a result of sytrf. It is internally computed by sytrf if omitted.
work : default = an optimum-length vector.
val potrf : ?up:[< `L | `U ] Slap_common.uplo ->
?jitter:num_type -> ('n, 'n, 'cd) mat -> unit
potrf ?up ?jitter a computes the Cholesky factorization of symmetrix (Hermitian) positive-definite matrix a:

where U and L are upper and lower triangular matrices, respectively. Either of them is returned in the upper or lower triangular part of a, as specified by up.
Raises Failure if a is not positive-definite symmetric.
up : default = Slap_common.upper
jitter : default = nothing
val potrs : ?up:[< `L | `U ] Slap_common.uplo ->
('n, 'n, 'a_cd) mat ->
?factorize:bool ->
?jitter:num_type -> ('n, 'nrhs, 'b_cd) mat -> unit
potrf ?up a ?jitter b solves systems of linear equations a * x = b using the Cholesky factorization of symmetrix (Hermitian) positive-definite matrix a:

where U and L are upper and lower triangular matrices, respectively.
Raises Failure if a is singular.
up : default = Slap_common.upper
factorize : default = true (potrf is called implicitly)
jitter : default = nothing
val potri : ?up:[< `L | `U ] Slap_common.uplo ->
?factorize:bool ->
?jitter:num_type -> ('n, 'n, 'cd) mat -> unit
potrf ?up ?jitter a computes the inverse of symmetrix (Hermitian) positive-definite matrix a using the Cholesky factorization:

where U and L are upper and lower triangular matrices, respectively.
Raises Failure if a is singular.
up : default = Slap_common.upper
factorize : default = true (potrf is called implicitly)
jitter : default = nothing
val trtrs : ?up:[< `L | `U ] Slap_common.uplo ->
trans:('n * 'n, 'n * 'n, [< `N | `T ]) trans3 ->
?diag:Slap_common.diag ->
('n, 'n, 'a_cd) mat -> ('n, 'nrhs, 'b_cd) mat -> unit
trtrs ?up trans ?diag a b solves systems of linear equations OP(a) * x = b where a is a triangular matrix of order 'n, each column of matrix b is the r.h.s vector, and each column of matrix x is the corresponding solution. The solution x is returned in b.
Raises Failure if a is singular.
up : default = Slap_common.upper
trans : the transpose flag for a:
diag : default = `N
val tbtrs : kd:'kd Slap_size.t ->
?up:[< `L | `U ] Slap_common.uplo ->
trans:('n * 'n, 'n * 'n, [< `N | `T ]) trans3 ->
?diag:Slap_common.diag ->
(('n, 'kd) Slap_size.syband, 'n, 'a_cd) mat ->
('n, 'nrhs, 'b_cd) mat -> unit
tbtrs ~kd ?up ~trans ?diag ab b solves systems of linear equations OP(A) * x = b where A is a triangular band matrix with kd subdiagonals, each column of matrix b is the r.h.s vector, and each column of matrix x is the corresponding solution. Matrix A is stored into ab in band storage format. The solution x is returned in b.
Since 0.2.0
Raises Failure if the matrix A is singular.
kd : the number of subdiagonals or superdiagonals in A.
up : default = Slap_common.upper
trans : the transpose flag for A:
diag : default = `N
val trtri : ?up:[< `L | `U ] Slap_common.uplo ->
?diag:Slap_common.diag -> ('n, 'n, 'cd) mat -> unit
trtri ?up ?diag a computes the inverse of triangular matrix a. The inverse matrix is returned in a.
Since 0.2.0
Raises Failure if the matrix a is singular.
up : default = Slap_common.upper
diag : default = `N
type 'n geqrf_min_lwork 
val geqrf_min_lwork : n:'n Slap_size.t -> 'n geqrf_min_lwork Slap_size.t
geqrf_min_lwork ~n computes the minimum length of workspace for geqrf routine. n is the number of columns in a matrix.
val geqrf_opt_lwork : ('m, 'n, 'cd) mat -> (module Slap_size.SIZE)
geqrf_opt_lwork a computes the optimum length of workspace for geqrf routine.
val geqrf : ?work:('lwork, Slap_misc.cnt) vec ->
?tau:(('m, 'n) Slap_size.min, Slap_misc.cnt) vec ->
('m, 'n, 'cd) mat -> (('m, 'n) Slap_size.min, 'cnt) vec
geqrf ?work ?tau a computes the QR factorization of general matrix a: a = Q * R where Q is an orthogonal (unitary) matrix and R is an upper triangular matrix. R is returned in a. This routine does not generate Q explicitly. It is generated by orgqr.
Returns vector tau, which is overwritten.
work : default = an optimum-length vector.

Linear equations (simple drivers)


val gesv : ?ipiv:('n, Slap_misc.cnt) Slap_common.int32_vec ->
('n, 'n, 'a_cd) mat -> ('n, 'nrhs, 'b_cd) mat -> unit
gesv ?ipiv a b solves a system of linear equations a * x = b where a is a 'n-by-'n general matrix, each column of matrix b is the r.h.s. vector, and each column of matrix x is the corresponding solution.

This routine uses LU factorization: a = P * L * U with permutation matrix P, a lower triangular matrix L and an upper triangular matrix U. By this function, the upper triangular part of a is replaced by U, the lower triangular part by L, and the solution x is returned in b.
Since 0.2.0
Raises Failure if the matrix is singular.

val gbsv : ?ipiv:('n, Slap_misc.cnt) Slap_common.int32_vec ->
(('n, 'n, 'kl, 'ku) Slap_size.luband, 'n, 'a_cd) mat ->
'kl Slap_size.t -> 'ku Slap_size.t -> ('n, 'nrhs, 'b_cd) mat -> unit
gbsv ?ipiv ab kl ku b solves a system of linear equations A * X = B where A is a 'n-by-'n band matrix, each column of matrix B is the r.h.s. vector, and each column of matrix X is the corresponding solution. The matrix A with kl subdiagonals and ku superdiagonals is stored into ab in band storage format for LU factorizaion.

This routine uses LU factorization: A = P * L * U with permutation matrix P, a lower triangular matrix L and an upper triangular matrix U. By this function, the upper triangular part of A is replaced by U, the lower triangular part by L, and the solution X is returned in B.
Since 0.2.0
Raises Failure if the matrix is singular.

val posv : ?up:[< `L | `U ] Slap_common.uplo ->
('n, 'n, 'a_cd) mat -> ('n, 'nrhs, 'b_cd) mat -> unit
posv ?up a b solves systems of linear equations a * x = b where a is a 'n-by-'n symmetric positive-definite matrix, each column of matrix b is the r.h.s vector, and each column of matrix x is the corresponding solution. The solution x is returned in b.

The Cholesky decomposition is used:

where U and L are the upper and lower triangular matrices, respectively.
Since 0.2.0
Raises Failure if the matrix is singular.
up : default = Slap_common.upper
val ppsv : ?up:[< `L | `U ] Slap_common.uplo ->
('n Slap_size.packed, Slap_misc.cnt) vec ->
('n, 'nrhs, 'b_cd) mat -> unit
ppsv ?up a b solves systems of linear equations a * x = b where a is a 'n-by-'n symmetric positive-definite matrix stored in packed format, each column of matrix b is the r.h.s vector, and each column of matrix x is the corresponding solution. The solution x is returned in b.

where U and L are the upper and lower triangular matrices, respectively.
Since 0.2.0
Raises Failure if the matrix is singular.
up : default = Slap_common.upper
val pbsv : ?up:[< `L | `U ] Slap_common.uplo ->
kd:'kd Slap_size.t ->
(('n, 'kd) Slap_size.syband, 'n, 'ab_cd) mat ->
('n, 'nrhs, 'b_cd) mat -> unit
pbsv ?up ~kd ab b solves systems of linear equations ab * x = b where ab is a 'n-by-'n symmetric positive-definite band matrix with kd subdiangonals, stored in band storage format, each column of matrix b is the r.h.s vector, and each column of matrix x is the corresponding solution. The solution x is returned in b.

This routine uses the Cholesky decomposition:

where U and L are the upper and lower triangular matrices, respectively.
Since 0.2.0
Raises Failure if the matrix is singular.
up : default = Slap_common.upper
kd : the number of subdiagonals or superdiagonals in ab.
val ptsv : ('n, Slap_misc.cnt) vec ->
('n Slap_size.p, Slap_misc.cnt) vec ->
('n, 'nrhs, 'b_cd) mat -> unit
ptsv d e b solves systems of linear equations A * x = b where A is a 'n-by-'n symmetric positive-definite tridiagonal matrix with diagonal elements d and subdiagonal elements e, each column of matrix b is the r.h.s vector, and each column of matrix x is the corresponding solution. The solution x is returned in b.

This routine uses the Cholesky decomposition: A = L^T * L (real) or A = L^H * L (complex) where L is a lower triangular matrix.
Since 0.2.0
Raises Failure if the matrix is singular.

type sysv_min_lwork 
val sysv_min_lwork : unit -> sysv_min_lwork Slap_size.t
sysv_min_lwork () computes the minimum length of workspace for sysv routine.
val sysv_opt_lwork : ?up:[< `L | `U ] Slap_common.uplo ->
('n, 'n, 'a_cd) mat ->
('n, 'nrhs, 'b_cd) mat -> (module Slap_size.SIZE)
sysv_opt_lwork ?up a b computes the optimal length of workspace for sysv routine.
val sysv : ?up:[< `L | `U ] Slap_common.uplo ->
?ipiv:('n, Slap_misc.cnt) Slap_common.int32_vec ->
?work:('lwork, Slap_misc.cnt) vec ->
('n, 'n, 'a_cd) mat -> ('n, 'nrhs, 'b_cd) mat -> unit
sysv ?up ?ipiv ?work a b solves systems of linear equations a * x = b where a is a 'n-by-'n symmetric matrix, each column of matrix b is the r.h.s. vector, and each column of matrix x is the corresponding solution. The solution x is returned in b.

The diagonal pivoting method is used:


Since 0.2.0
Raises Failure if the matrix is singular.
up : default = Slap_common.upper
ipiv : a result of sytrf. It is internally computed by sytrf if omitted.
work : default = an optimum-length vector.
val spsv : ?up:[< `L | `U ] Slap_common.uplo ->
?ipiv:('n, Slap_misc.cnt) Slap_common.int32_vec ->
('n Slap_size.packed, Slap_misc.cnt) vec ->
('n, 'nrhs, 'b_cd) mat -> unit
spsv ?up a b solves systems of linear equations a * x = b where a is a 'n-by-'n symmetric matrix stored in packed format, each column of matrix b is the r.h.s. vector, and each column of matrix x is the corresponding solution. The solution x is returned in b.

The diagonal pivoting method is used:


Since 0.2.0
Raises Failure if the matrix is singular.
up : default = Slap_common.upper
ipiv : a result of sytrf. It is internally computed by sytrf if omitted.

Least squares (simple drivers)


type ('m, 'n, 'nrhs) gels_min_lwork 
val gels_min_lwork : m:'m Slap_size.t ->
n:'n Slap_size.t ->
nrhs:'nrhs Slap_size.t -> ('m, 'n, 'nrhs) gels_min_lwork Slap_size.t
gels_min_lwork ~n computes the minimum length of workspace for gels routine.
m : the number of rows in a matrix.
n : the number of columns in a matrix.
nrhs : the number of right hand sides.
val gels_opt_lwork : trans:('am * 'an, 'm * 'n, [< `N | `T ]) Slap_common.trans2 ->
('am, 'an, 'a_cd) mat ->
('m, 'nrhs, 'b_cd) mat -> (module Slap_size.SIZE)
gels_opt_lwork ~trans a b computes the optimum length of workspace for gels routine.
val gels : ?work:('work, Slap_misc.cnt) vec ->
trans:('am * 'an, 'm * 'n, [< `N | `T ]) Slap_common.trans2 ->
('am, 'an, 'a_cd) mat -> ('m, 'nrhs, 'b_cd) mat -> unit
gels ?work ~trans a b solves an overdetermined or underdetermined system of linear equations using QR or LU factorization.


work : default = an optimum-length vector.
trans : the transpose flag for a.

BLAS interface



BLAS interface



Level 1


val dot : ('n, 'x_cd) vec -> ('n, 'y_cd) vec -> float
dot x y
Returns the inner product of the vectors x and y.
val asum : ('n, 'x_cd) vec -> float
asum x
Returns the sum of absolute values of elements in the vector x.

Level 2


val sbmv : k:'k Slap_size.t ->
?y:('n, 'y_cd) vec ->
(('n, 'k) Slap_size.syband, 'n, 'a_cd) mat ->
?up:[< `L | `U ] Slap_common.uplo ->
?alpha:float ->
?beta:float -> ('n, 'x_cd) vec -> ('n, 'y_cd) vec
sbmv ~k ?y a ?up ?alpha ?beta x computes y := alpha * a * x + beta * y where a is a 'n-by-'n symmetric band matrix with k super-(or sub-)diagonals, and x and y are 'n-dimensional vectors.
Since 0.2.0
Returns vector y, which is overwritten.
k : the number of superdiangonals or subdiangonals
up : default = Slap_common.upper
alpha : default = 1.0
beta : default = 0.0
val ger : ?alpha:float ->
('m, 'x_cd) vec ->
('n, 'y_cd) vec ->
('m, 'n, 'a_cd) mat -> ('m, 'n, 'a_cd) mat
ger ?alpha x y a computes a := alpha * x * y^T + a with the general matrix a, the vector x and the transposed vector y^T of y.
Returns matrix a, which is overwritten.
alpha : default = 1.0
val syr : ?alpha:float ->
?up:[< `L | `U ] Slap_common.uplo ->
('n, 'x_cd) vec ->
('n, 'n, 'a_cd) mat -> ('n, 'n, 'a_cd) mat
syr ?alpha x a computes a := alpha * x * x^T + a with the symmetric matrix a, the vector x and the transposed vector x^T of x.
Returns matrix a, which is overwritten.
alpha : default = 1.0
up : default = Slap_common.upper

LAPACK interface



LAPACK interface



Auxiliary routines



LAPACK interface



Auxiliary routines



lansy


type ('m, 'a) lansy_min_lwork 
val lansy_min_lwork : 'n Slap_size.t ->
([< `F | `I | `M | `O ] as 'a) Slap_common.norm4 ->
('n, 'a) lansy_min_lwork Slap_size.t
lansy_min_lwork n norm computes the minimum length of workspace for lansy routine. n is the number of rows or columns in a matrix. norm is a matrix norm.
val lansy : ?up:[< `L | `U ] Slap_common.uplo ->
?norm:[< `F | `I | `M | `O ] Slap_common.norm4 ->
?work:('lwork, Slap_misc.cnt) vec -> ('n, 'n, 'cd) mat -> float
lansy ?up ?norm ?work a
Returns the norm of matrix a.
up : default = Slap_common.upper
norm : default = Slap_common.norm_1
work : default = an optimum-length vector.

lamch


val lamch : [ `B | `E | `L | `M | `N | `O | `P | `R | `S | `U ] -> float
lamch cmach see LAPACK documentation.

Linear equations (computational routines)



Linear equations (computational routines)



orgqr


type 'n orgqr_min_lwork 
val orgqr_min_lwork : n:'n Slap_size.t -> 'n orgqr_min_lwork Slap_size.t
orgqr_min_lwork ~n computes the minimum length of workspace for orgqr routine. n is the number of columns in a matrix.
val orgqr_opt_lwork : tau:('k, Slap_misc.cnt) vec ->
('m, 'n, 'cd) mat -> (module Slap_size.SIZE)
orgqr_min_lwork ~tau a computes the optimum length of workspace for orgqr routine.
val orgqr_dyn : ?work:('lwork, Slap_misc.cnt) vec ->
tau:('k, Slap_misc.cnt) vec -> ('m, 'n, 'cd) mat -> unit
orgqr_dyn ?work ~tau a generates the orthogonal matrix Q of the QR factorization formed by geqrf/geqpf.
Raises Invalid_argument if the following inequality is not satisfied: (Mat.dim1 a) >= (Mat.dim2 a) >= (Vec.dim tau), i.e., 'm >= 'n >= 'k.
work : default = an optimum-length vector
tau : a result of geqrf

ormqr


type ('r, 'm, 'n) ormqr_min_lwork 
val ormqr_min_lwork : side:('r, 'm, 'n) Slap_common.side ->
m:'m Slap_size.t ->
n:'n Slap_size.t -> ('r, 'm, 'n) ormqr_min_lwork Slap_size.t
ormqr_min_lwork ~side ~m ~n computes the minimum length of workspace for ormqr routine.
side : the side flag to specify direction of matrix multiplication.
m : the number of rows in a matrix.
n : the number of columns in a matrix.
val ormqr_opt_lwork : side:('r, 'm, 'n) Slap_common.side ->
trans:('r * 'r, 'r * 'r, [< `N | `T ]) Slap_common.trans2 ->
tau:('k, Slap_misc.cnt) vec ->
('r, 'k, 'a_cd) mat ->
('m, 'n, 'c_cd) mat -> (module Slap_size.SIZE)
ormqr_opt_lwork ~side ~trans ~tau a c computes the optimum length of workspace for ormqr routine.
side : the side flag to specify direction of matrix multiplication.
trans : the transpose flag for orthogonal matrix Q.
tau : a result of geqrf.
val ormqr_dyn : side:('r, 'm, 'n) Slap_common.side ->
trans:('r * 'r, 'r * 'r, [< `N | `T ]) Slap_common.trans2 ->
?work:('lwork, Slap_misc.cnt) vec ->
tau:('k, Slap_misc.cnt) vec ->
('r, 'k, 'a_cd) mat -> ('m, 'n, 'c_cd) mat -> unit
ormqr_dyn ~side ~trans ?work ~tau a c multiplies a matrix c by the orthogonal matrix Q of the QR factorization formed by geqrf/geqpf:


Raises Invalid_argument if the following inequality is not satisfied:
side : the side flag to specify direction of matrix multiplication of Q and c.
trans : the transpose flag for orthogonal matrix Q.
work : default = an optimum-length vector.
tau : a result of geqrf.

gecon


type 'n gecon_min_lwork 
val gecon_min_lwork : 'n Slap_size.t -> 'n gecon_min_lwork Slap_size.t
gecon_min_lwork n computes the minimum length of workspace work for gecon routine. n is the number of rows or columns in a matrix.
type 'n gecon_min_liwork 
val gecon_min_liwork : 'n Slap_size.t -> 'n gecon_min_liwork Slap_size.t
gecon_min_liwork n computes the minimum length of workspace iwork for gecon routine. n is the number of rows or columns in a matrix.
val gecon : ?norm:[< `I | `O ] Slap_common.norm2 ->
?anorm:float ->
?work:('lwork, Slap_misc.cnt) vec ->
?iwork:('liwork, Slap_misc.cnt) Slap_common.int32_vec ->
('n, 'n, 'cd) mat -> float
gecon ?norm ?anorm ?work ?iwork a estimates the reciprocal of the condition number of general matrix a.
norm : default = Slap_common.norm_1.
anorm : default = the norm of matrix a as returned by lange.
work : default = an optimum-length vector.
iwork : default = an optimum-length vector.

sycon


type 'n sycon_min_lwork 
val sycon_min_lwork : 'n Slap_size.t -> 'n sycon_min_lwork Slap_size.t
sycon_min_lwork n computes the minimum length of workspace work for sycon routine. n is the number of rows or columns in a matrix.
type 'n sycon_min_liwork 
val sycon_min_liwork : 'n Slap_size.t -> 'n sycon_min_liwork Slap_size.t
sycon_min_liwork n computes the minimum length of workspace iwork for sycon routine. n is the number of rows or columns in a matrix.
val sycon : ?up:[< `L | `U ] Slap_common.uplo ->
?ipiv:('n, Slap_misc.cnt) Slap_common.int32_vec ->
?anorm:float ->
?work:('lwork, Slap_misc.cnt) vec ->
?iwork:('liwork, Slap_misc.cnt) Slap_common.int32_vec ->
('n, 'n, 'cd) mat -> float
sycon ?up ?ipiv ?anorm ?work ?iwork a estimates the reciprocal of the condition number of symmetric matrix a. Since a is symmetric, the 1-norm is equal to the infinity norm.
up : default = Slap_common.upper
ipiv : a result of sytrf. It is internally computed by sytrf if omitted.
anorm : default = the norm of matrix a as returned by lange.
work : default = an optimum-length vector.
iwork : default = an optimum-length vector.

pocon


type 'n pocon_min_lwork 
val pocon_min_lwork : 'n Slap_size.t -> 'n pocon_min_lwork Slap_size.t
pocon_min_lwork n computes the minimum length of workspace work for pocon routine. n is the number of rows or columns in a matrix.
type 'n pocon_min_liwork 
val pocon_min_liwork : 'n Slap_size.t -> 'n pocon_min_liwork Slap_size.t
pocon_min_liwork n computes the minimum length of workspace iwork for pocon routine. n is the number of rows or columns in a matrix.
val pocon : ?up:[< `L | `U ] Slap_common.uplo ->
?anorm:float ->
?work:('lwork, Slap_misc.cnt) vec ->
?iwork:('liwork, Slap_misc.cnt) Slap_common.int32_vec ->
('n, 'n, 'cd) mat -> float
pocon ?up ?anorm ?work ?iwork a estimates the reciprocal of the condition number of symmetric positive-definite matrix a. Since a is symmetric, the 1-norm is equal to the infinity norm.
up : default = Slap_common.upper
anorm : default = the norm of matrix a as returned by lange.
work : default = an optimum-length vector.
iwork : default = an optimum-length vector.

Least squares (expert drivers)



Least squares (expert drivers)



gelsy


type ('m, 'n, 'nrhs) gelsy_min_lwork 
val gelsy_min_lwork : m:'m Slap_size.t ->
n:'n Slap_size.t ->
nrhs:'nrhs Slap_size.t -> ('m, 'n, 'nrhs) gelsy_min_lwork Slap_size.t
gelsy_min_lwork ~m ~n ~nrhs computes the minimum length of workspace for gelsy routine.
m : the number of rows in a matrix.
n : the number of columns in a matrix.
nrhs : the number of right hand sides.
val gelsy_opt_lwork : ('m, 'n, 'a_cd) mat ->
('n, 'nrhs, 'b_cd) mat -> (module Slap_size.SIZE)
gelsy_opt_lwork a b computes the optimum length of workspace for gelsy routine.
val gelsy : ('m, 'n, 'a_cd) mat ->
?rcond:float ->
?jpvt:('n, Slap_misc.cnt) Slap_common.int32_vec ->
?work:('lwork, Slap_misc.cnt) vec ->
('n, 'nrhs, 'b_cd) mat -> int
gelsy a ?rcond ?jpvt ?work b computes the minimum-norm solution to a linear least square problem (minimize ||b - a * x||) using a complete orthogonal factorization of a.
Returns the effective rank of a.
rcond : default = -1.0 (machine precision)
jpvt : default = a 'n-dimensional vector.
work : default = an optimum-length vector.

gelsd


type ('m, 'n, 'nrhs) gelsd_min_lwork 
val gelsd_min_lwork : m:'m Slap_size.t ->
n:'n Slap_size.t ->
nrhs:'nrhs Slap_size.t -> ('m, 'n, 'nrhs) gelsd_min_lwork Slap_size.t
gelsd_min_lwork ~m ~n ~nrhs computes the minimum length of workspace for gelsd routine.
m : the number of rows in a matrix.
n : the number of columns in a matrix.
nrhs : the number of right hand sides.
val gelsd_opt_lwork : ('m, 'n, 'a_cd) mat ->
('n, 'nrhs, 'b_cd) mat -> (module Slap_size.SIZE)
gelsd_opt_lwork a b computes the optimum length of workspace for gelsd routine.
type ('m, 'n, 'nrhs) gelsd_min_iwork 
val gelsd_min_iwork : 'm Slap_size.t ->
'n Slap_size.t -> ('m, 'n, 'nrhs) gelsd_min_iwork Slap_size.t
gelsd_min_iwork ~m ~n ~nrhs computes the minimum length of workspace iwork for gelsd routine.
val gelsd : ('m, 'n, 'a_cd) mat ->
?rcond:float ->
?s:(('m, 'n) Slap_size.min, Slap_misc.cnt) vec ->
?work:('lwork, Slap_misc.cnt) vec ->
?iwork:('liwork, Slap_misc.cnt) vec ->
('n, 'nrhs, 'b_cd) mat -> int
gelsd a ?rcond ?jpvt ?work b computes the minimum-norm solution to a linear least square problem (minimize ||b - a * x||) using the singular value decomposition (SVD) of a and a divide and conquer method.
Raises Failure if the function fails to converge.
Returns the effective rank of a.
rcond : default = -1.0 (machine precision)
s : the singular values of a in decreasing order. They are implicitly computed if omitted.
work : default = an optimum-length vector.
iwork : default = an optimum-length vector.

gelss


type ('m, 'n, 'nrhs) gelss_min_lwork 
val gelss_min_lwork : m:'m Slap_size.t ->
n:'n Slap_size.t ->
nrhs:'nrhs Slap_size.t -> ('m, 'n, 'nrhs) gelss_min_lwork Slap_size.t
gelss_min_lwork ~m ~n ~nrhs computes the minimum length of workspace for gelss routine.
m : the number of rows in a matrix.
n : the number of columns in a matrix.
nrhs : the number of right hand sides.
val gelss_opt_lwork : ('m, 'n, 'a_cd) mat ->
('n, 'nrhs, 'b_cd) mat -> (module Slap_size.SIZE)
gelss_min_lwork a b computes the optimum length of workspace for gelss routine.
val gelss : ('m, 'n, 'a_cd) mat ->
?rcond:float ->
?s:(('m, 'n) Slap_size.min, Slap_misc.cnt) vec ->
?work:('lwork, Slap_misc.cnt) vec ->
('n, 'nrhs, 'b_cd) mat -> int
gelss a ?rcond ?work b computes the minimum-norm solution to a linear least square problem (minimize ||b - a * x||) using the singular value decomposition (SVD) of a.
Raises Failure if the function fails to converge.
Returns the effective rank of a.
rcond : default = -1.0 (machine precision)
s : the singular values of a in decreasing order. They are implicitly computed if omitted.
work : default = an optimum-length vector.

General SVD routines



General SVD routines



gesvd


type ('m, 'n) gesvd_min_lwork 
val gesvd_min_lwork : m:'m Slap_size.t ->
n:'n Slap_size.t -> ('m, 'n) gesvd_min_lwork Slap_size.t
gesvd_min_lwork ~m ~n computes the minimum length of workspace for gesvd routine.
m : the number of rows in a matrix.
n : the number of columns in a matrix.
val gesvd_opt_lwork : jobu:('u_cols, 'm, ('m, 'n) Slap_size.min, Slap_size.z, Slap_size.z)
Slap_common.svd_job ->
jobvt:('vt_rows, 'n, ('m, 'n) Slap_size.min, Slap_size.z, Slap_size.z)
Slap_common.svd_job ->
?s:(('m, 'n) Slap_size.min, Slap_misc.cnt) vec ->
?u:('m, 'u_cols, 'u_cd) mat ->
?vt:('vt_rows, 'n, 'vt_cd) mat ->
('m, 'n, 'a_cd) mat -> (module Slap_size.SIZE)
gesvd_opt_lwork ~jobu ~jobvt ?s ?u ?vt computes the optimum length of workspace for gesvd routine.
jobu : the SVD job flag for u.
jobvt : the SVD job flag for vt.
s : a return location for singular values.
u : a return location for left singular vectors.
vt : a return location for (transposed) right singular vectors.
val gesvd : jobu:('u_cols, 'm, ('m, 'n) Slap_size.min, Slap_size.z, Slap_size.z)
Slap_common.svd_job ->
jobvt:('vt_rows, 'n, ('m, 'n) Slap_size.min, Slap_size.z, Slap_size.z)
Slap_common.svd_job ->
?s:(('m, 'n) Slap_size.min, Slap_misc.cnt) vec ->
?u:('m, 'u_cols, 'u_cd) mat ->
?vt:('vt_rows, 'n, 'vt_cd) mat ->
?work:('lwork, Slap_misc.cnt) vec ->
('m, 'n, 'a_cd) mat ->
(('m, 'n) Slap_size.min, 'cnt) vec * ('m, 'u_cols, 'cnt) mat *
('vt_rows, 'n, 'cnt) mat
gesvd ?jobu ?jobvt ?s ?u ?vt ?work a computes the singular value decomposition (SVD) of 'm-by-'n general rectangular matrix a: a = U * D * V^T where


Returns (s, u, vt) with singular values s in descreasing order, left singular vectors u and right singular vectors vt.
jobu : the SVD job flag for u:
jobvt : the SVD job flag for vt:
s : a return location for singular values. (default = an implicitly allocated vector.)
u : a return location for left singular vectors. (default = an implicitly allocated matrix.)
vt : a return location for (transposed) right singular vectors. (default = an implicitly allocated matrix.)
work : default = an optimum-length vector.

(Note: jobu and jobvt cannot both be Slap_common.svd_overwrite.)


gesdd


type ('m, 'n) gesdd_liwork 
val gesdd_liwork : m:'m Slap_size.t ->
n:'n Slap_size.t -> ('m, 'n) gesdd_liwork Slap_size.t
gesdd_liwork ~m ~n computes the length of workspace iwork for gesdd routine.
m : the number of rows in a matrix.
n : the number of columns in a matrix.
type ('m, 'n, 'jobz) gesdd_min_lwork 
val gesdd_min_lwork : jobz:('u_cols * 'vt_rows, 'm * 'n,
('m, 'n) Slap_size.min * ('m, 'n) Slap_size.min, 'm * 'n,
Slap_size.z * Slap_size.z)
Slap_common.svd_job ->
m:'m Slap_size.t ->
n:'n Slap_size.t ->
unit -> ('m, 'n, 'u_cols * 'vt_rows) gesdd_min_lwork Slap_size.t
gesdd_min_lwork ~m ~n computes the minimum length of workspace work for gesdd routine.
jobz : the SVD job flag.
m : the number of rows in a matrix.
n : the number of columns in a matrix.
val gesdd_opt_lwork : jobz:('u_cols * 'vt_rows, 'm * 'n,
('m, 'n) Slap_size.min * ('m, 'n) Slap_size.min, 'm * 'n,
Slap_size.z * Slap_size.z)
Slap_common.svd_job ->
?s:(('m, 'n) Slap_size.min, Slap_misc.cnt) vec ->
?u:('m, 'u_cols, 'u_cd) mat ->
?vt:('vt_rows, 'n, 'vt_cd) mat ->
?iwork:('liwork, Slap_misc.cnt) Slap_common.int32_vec ->
('m, 'n, 'a_cd) mat -> (module Slap_size.SIZE)
gesdd_opt_lwork ~jobz ?s ?u ?vt ?iwork a computes the optimum length of workspace work for gesdd routine.
jobz : the SVD job flag.
s : a return location for singular values. (default = an implicitly allocated vector.)
u : a return location for left singular vectors. (default = an implicitly allocated matrix.)
vt : a return location for (transposed) right singular vectors. (default = an implicitly allocated matrix.)
iwork : default = an optimum-length vector.
val gesdd : jobz:('u_cols * 'vt_rows, 'm * 'n,
('m, 'n) Slap_size.min * ('m, 'n) Slap_size.min, 'm * 'n,
Slap_size.z * Slap_size.z)
Slap_common.svd_job ->
?s:(('m, 'n) Slap_size.min, Slap_misc.cnt) vec ->
?u:('m, 'u_cols, 'u_cd) mat ->
?vt:('vt_rows, 'n, 'vt_cd) mat ->
?work:('lwork, Slap_misc.cnt) vec ->
?iwork:('liwork, Slap_misc.cnt) Slap_common.int32_vec ->
('m, 'n, 'a_cd) mat ->
(('m, 'n) Slap_size.min, 'cnt) vec *
('m, 'u_cols, 'u_cd) mat option *
('vt_rows, 'n, 'vt_cd) mat option
gesdd ~jobz ?s ?u ?vt ?work ?iwork a computes the singular value decomposition (SVD) of general rectangular matrix a using a divide and conquer method: a = U * D * V^T where


Returns (s, u, vt) with singular values s in descreasing order, left singular vectors u and right singular vectors vt. If u (vt) is not needed, None is returned.
jobz : the SVD job flag:
s : a return location for singular values. (default = an implicitly allocated vector.)
u : a return location for left singular vectors. (default = an implicitly allocated matrix.)
vt : a return location for (transposed) right singular vectors. (default = an implicitly allocated matrix.)
work : default = an optimum-length vector.
iwork : default = an optimum-length vector.

General eigenvalue problem (simple drivers)



General eigenvalue problem (simple drivers)



geev


val geev_min_lwork : ?vectors:bool -> 'n Slap_size.t -> (module Slap_size.SIZE)
geev_min_lwork ?vectors n computes the minimum length of workspace for geev routine. n is the number of rows or columns of a matrix.
vectors : whether eigenvectors are computed, or not. (default = true, i.e., they are computed.)
val geev_opt_lwork : ?vl:('n, 'n, 'vl_cd) mat option ->
?vr:('n, 'n, 'vr_cd) mat option ->
?wr:('n, Slap_misc.cnt) vec ->
?wi:('n, Slap_misc.cnt) vec ->
('n, 'n, 'a_cd) mat -> (module Slap_size.SIZE)
geev_opt_lwork ?vl ?vr ?wr ?vi a computes the optimum length of workspace for geev routine.
vl : a return location for left eigenvectors. See LAPACK GEEV documentation for details about storage of complex eigenvectors. (default = an implicitly allocated matrix.)
vr : a return location for right eigenvectors. See LAPACK GEEV documentation for details about storage of complex eigenvectors. (default = an implicitly allocated matrix.)
wr : a return location for real parts of eigenvalues. (default = an implicitly allocated vector.)
wi : a return location for imaginary parts of eigenvalues. (default = an implicitly allocated vector.)
val geev : ?work:('lwork, Slap_misc.cnt) vec ->
?vl:('n, 'n, 'vl_cd) mat option ->
?vr:('n, 'n, 'vr_cd) mat option ->
?wr:('n, Slap_misc.cnt) vec ->
?wi:('n, Slap_misc.cnt) vec ->
('n, 'n, 'a_cd) mat ->
('n, 'n, 'vl_cd) mat option * ('n, Slap_misc.cnt) vec *
('n, Slap_misc.cnt) vec * ('n, 'n, 'vr_cd) mat option
geev ?work ?vl ?vr ?wr ?wi a computes the eigenvalues and the left and right eigenvectors of 'n-by-'n nonsymmetric matrix a:

Let w(j) is the j-th eigenvalue of a. The j-th right eigenvector vr(j) satisfies a * vr(j) = w(j) * vr(j), and the j-th left eigenvector vl(j) satisfies vl(j)^H * a = vl(j)^H * w(j) where vl(j)^H denotes the conjugate transpose of vl(j). The computed eigenvalues are normalized by Euclidian norm.
Raises Failure if the function fails to converge.
Returns (vl, wr, wi, vr) where vl and vr are left and right eigenvectors, and wr and wi are the real and imaginary parts of eigenvalues. If vl (vr) is an empty matrix, None is set.

work : default = an optimum-length vector.
vl : a return location for left eigenvectors. See LAPACK GEEV documentation for details about storage of complex eigenvectors. (default = an implicitly allocated matrix.)
vr : a return location for right eigenvectors. See LAPACK GEEV documentation for details about storage of complex eigenvectors. (default = an implicitly allocated matrix.)
wr : a return location for real parts of eigenvalues. (default = an implicitly allocated vector.)
wi : a return location for imaginary parts of eigenvalues. (default = an implicitly allocated vector.)

Symmetric-matrix eigenvalue and singular value problems (simple drivers)



Symmetric-matrix eigenvalue and singular value problems (simple drivers)



syev


type 'n syev_min_lwork 
val syev_min_lwork : 'n Slap_size.t -> 'n syev_min_lwork Slap_size.t
syev_min_lwork n computes the minimum length of workspace for syev routine. n is the number of rows or columns of a matrix.
val syev_opt_lwork : ?vectors:bool ->
?up:bool -> ('n, 'n, 'cd) mat -> (module Slap_size.SIZE)
syev_opt_lwork ?vectors ?up a computes the optimum length of workspace for syev routine.
vectors : whether eigenvectors are computed, or not. (default = false.)
up : default = true
val syev : ?vectors:bool ->
?up:bool ->
?work:('lwork, Slap_misc.cnt) vec ->
?w:('n, Slap_misc.cnt) vec ->
('n, 'n, 'cd) mat -> ('n, 'cnt) vec
syev ?vectors ?up ?work ?w a computes the eigenvalues and the eigenvectors of 'n-by-'n symmetric matrix a.
Raises Failure if the function fails to converge.
Returns the vector w of eigenvalues in ascending order.
vectors : whether eigenvectors are computed, or not. (default = false.)
up : default = true
work : default = an optimum-length vector.
w : a return location for eigenvalues. (default = an implicitly allocated vector.)

syevd


val syevd_min_lwork : vectors:bool -> 'n Slap_size.t -> (module Slap_size.SIZE)
syevd_min_lwork ?vectors n computes the minimum length of workspace work for syevd routine. n is the number of rows or columns of a matrix.
vectors : whether eigenvectors are computed, or not. (default = false, i.e., they are not computed.)
val syevd_min_liwork : vectors:bool -> 'n Slap_size.t -> (module Slap_size.SIZE)
syevd_min_liwork ?vectors n computes the minimum length of workspace iwork for syevd routine. n is the number of rows or columns of a matrix.
vectors : whether eigenvectors are computed, or not. (default = false, i.e., they are not computed.)
val syevd_opt_lwork : ?vectors:bool ->
?up:bool -> ('n, 'n, 'cd) mat -> (module Slap_size.SIZE)
syevd_opt_lwork ?vectors ?up a computes the optimum length of workspace work for syevd routine.
vectors : whether eigenvectors are computed, or not. (default = false, i.e., they are not computed.)
up : default = true
val syevd_opt_liwork : ?vectors:bool ->
?up:bool -> ('n, 'n, 'cd) mat -> (module Slap_size.SIZE)
syevd_opt_liwork ?vectors ?up a computes the optimum length of workspace iwork for syevd routine.
vectors : whether eigenvectors are computed, or not. (default = false, i.e., they are not computed.)
up : default = true
val syevd : ?vectors:bool ->
?up:bool ->
?work:('lwork, Slap_misc.cnt) vec ->
?iwork:('liwork, Slap_misc.cnt) Slap_common.int32_vec ->
?w:('n, Slap_misc.cnt) vec ->
('n, 'n, 'a_cd) mat -> ('n, 'w_cd) vec
syev ?vectors ?up ?w a computes the eigenvalues and the eigenvectors of 'n-by-'n symmetric matrix a using divide and conquer method.
Raises Failure if the function fails to converge.
Returns the vector w of eigenvalues in ascending order.
vectors : whether eigenvectors are computed, or not. (default = false.)
up : default = true
work : default = an optimum-length vector.
iwork : default = an optimum-length vector.
w : a return location for eigenvalues. (default = an implicitly allocated vector.)

sbev


type 'n sbev_min_lwork 
val sbev_min_lwork : 'n Slap_size.t -> 'n sbev_min_lwork Slap_size.t
sbev_min_lwork n computes the minimum length of workspace work for sbev routine. n is the number of rows or columns of a matrix.
val sbev : kd:'kd Slap_size.t ->
?z:('n, 'n, 'z_cd) mat ->
?up:bool ->
?work:('lwork, Slap_misc.cnt) vec ->
?w:('n, Slap_misc.cnt) vec ->
(('n, 'kd) Slap_size.syband, 'n, 'a_cd) mat -> ('n, 'cnt) vec
sbev ~kd ?z ?up ?work ?w ab computes all eigenvalues and, optionally, eigenvectors of real symmetric band matrix ab store in band storage format.
Since 0.2.0
Raises Failure if the function fails to converge.
Returns vector w, which is overwritten.
kd : the number of subdiagonals or superdiagonals.
z : The eigenvectors are returned in z if it is given. They are not computed if omitted.
up : default = true
work : workspace for sbev
w : w is replaced by eigenvalues if it is given, or newly allocated if omitted.

Symmetric-matrix eigenvalue and singular value problems (expert & RRR drivers)



Symmetric-matrix eigenvalue and singular value problems (expert & RRR drivers)



syevr


type 'n syevr_min_lwork 
val syevr_min_lwork : 'n Slap_size.t -> 'n syevr_min_lwork Slap_size.t
sbevr_min_lwork n computes the minimum length of workspace work for syevr routine. n is the number of rows or columns of a matrix.
type 'n syevr_min_liwork 
val syevr_min_liwork : 'n Slap_size.t -> 'n syevr_min_liwork Slap_size.t
sbevr_min_liwork n computes the minimum length of workspace iwork for syevr routine. n is the number of rows or columns of a matrix.
val syevr_opt_lwork : ?vectors:bool ->
?range:[ `A | `I of int * int | `V of float * float ] ->
?up:bool ->
?abstol:float -> ('n, 'n, 'a_cd) mat -> (module Slap_size.SIZE)
sbevr_opt_lwork ?vectors ?range ?up ?abstol a computes the optimum length of workspace work for syevr routine.
vectors : whether eigenvectors are computed, or not. (default = false, i.e., they are not computed.)
range : default = `A.
up : default = true.
abstol : The absolute error tolerance to which each eigenvalue or eigenvector is required. (default = lamch `S.)
val syevr_opt_liwork : ?vectors:bool ->
?range:[ `A | `I of int * int | `V of float * float ] ->
?up:bool ->
?abstol:float -> ('n, 'n, 'a_cd) mat -> (module Slap_size.SIZE)
sbevr_opt_liwork ?vectors ?range ?up ?abstol a computes the optimum length of workspace iwork for sbevr routine.
vectors : whether eigenvectors are computed, or not. (default = false, i.e., they are not computed.)
range : default = `A.
up : default = true.
abstol : The absolute error tolerance to which each eigenvalue or eigenvector is required. (default = lamch `S.)
module type SYEVR_RESULT = sig .. end
The signature of returned modules of syevr_dyn.
val syevr_dyn : ?vectors:bool ->
?range:[ `A | `I of int * int | `V of float * float ] ->
?up:bool ->
?abstol:float ->
?work:('lwork, Slap_misc.cnt) vec ->
?iwork:('liwork, Slap_misc.cnt) Slap_common.int32_vec ->
?w:('n, Slap_misc.cnt) vec ->
?z:('n, 'k, 'z_cd) mat ->
?isuppz:(('k, 'k) Slap_size.add, Slap_misc.cnt) Slap_common.int32_vec ->
('n, 'n, 'a_cd) mat -> (module Slap_D.SYEVR_RESULT with type n = 'n)
syevr_dyn ?vectors ?range ?up ?abstol ?work ?iwork ?w ?z ?isuppz a computes selected eigenvalues w and, optionally, eigenvectors z of a real symmetric matrix a using the Relatively Robust Representations.

Usage:

let f (type nn) ... =
  ...
  let (a : (nn, nn, _) mat) = ... in
  let module X = (val syevr_dyn ?vectors ?range ?up ?abstol
                      ?work ?iwork ?w ?z ?isuppz a
                   : SYEVR_RESULT with type n = nn) in
  let (m, w, z, isuppz) = X.value in
  ...
where type nn is the size of symmetric matrix a. The returned module X contains tuple X.value = (m, w, z, isuppz) and type X.m for representation of the number of computed eigenvalues:


Raises Invalid_argument if not X.m <= 'k
Returns the above-mentioned module X.
vectors : whether eigenvectors are computed, or not. (default = false.)
range : default = `A
up : default = true
abstol : The absolute error tolerance to which each eigenvalue or eigenvector is required. (default = lamch `S.)
work : default = an optimum-length vector.
iwork : default = an optimum-length vector.
w : a return location for eigenvalues. (default = an implicitly allocated vector of the minimum required dimension.)
z : a return location for eigenvectors. (default = an implicitly allocated matrix of the minimum required dimension.)
isuppz : a return location for a vector to indicate the nonzero elements in z. (default = an implicitly allocated matrix of the minimum required dimension.)

sygv


val sygv_opt_lwork : ?vectors:bool ->
?up:bool ->
?itype:[ `AB | `A_B | `BA ] ->
('n, 'n, 'a_cd) mat ->
('n, 'n, 'b_cd) mat -> (module Slap_size.SIZE)
sygv_opt_lwork ?vectors ?up ?itype a b computes the optimum length of workspace work for sbevr routine.
vectors : whether eigenvectors are computed, or not. (default = false, i.e., they are not computed.)
up : default = true.
itype : the behavior of this routine.
val sygv : ?vectors:bool ->
?up:bool ->
?work:('lwork, Slap_misc.cnt) vec ->
?w:('n, Slap_misc.cnt) vec ->
?itype:[ `AB | `A_B | `BA ] ->
('n, 'n, 'a_cd) mat ->
('n, 'n, 'b_cd) mat -> ('n, 'cnt) vec
sygv ?vectors ?up ?work ?w ?itype a b solves a real generalized symmetric definite eigenproblem:

where a is a 'n-by-'n symmetric matrix, and b is a 'n-by-'n symmetric positive definite matrix.
Raises Returns vector w of eigenvalues in ascending order.
vectors : whether eigenvectors are computed, or not. (default = false.)
up : default = true.
work : default = an optimum-length workspace.
w : a return location for eigenvalues. (default = an implicitly allocated vector.)
itype : the behavior of this routine.

sbgv


val sbgv : ka:'ka Slap_size.t ->
kb:'kb Slap_size.t ->
?z:('n, 'n, 'z_cd) mat ->
?up:bool ->
?work:('lwork, Slap_misc.cnt) vec ->
?w:('n, Slap_misc.cnt) vec ->
(('n, 'ka) Slap_size.syband, 'n, 'ab_cd) mat ->
(('n, 'kb) Slap_size.syband, 'n, 'bb_cd) mat -> ('n, 'cnt) vec
sbgv ~ka ~kb ?z ?up ?work ?w ab bb solves a general eigenvalue problem ab * z = (lambda) * bb * z where ab is a 'n-by-'n symmetric band matrix with ka subdiagonals, and bb is a 'n-by-'n symmetric positive-definite band matrix with kb subdiagonals. Both ab and bb are stored in band storage format.
Since 0.2.0
Raises Failure if the function fails to converge.
Returns vector w, which is overwritten.
ka : the number of subdiagonals or superdiagonals of ab.
kb : the number of subdiagonals or superdiagonals of bb.
z : The eigenvectors are returned in z if it is given. They are not computed if omitted.
up : default = true
work : workspace for sbgv
w : w is replaced by eigenvalues if it is given, or newly allocated if omitted.