Module Slap.Z

module Z: Slap_Z

64-bit complex BLAS and LAPACK functions.

type prec = Bigarray.complex64_elt

type num_type = Complex.t

type ('indim, 'outdim, [< `C | `N | `T ]) trans3 = ('indim, 'outdim, [< `C | `N | `T ] as 'a) Slap_common.trans3

A type of transpose parameters (Slap_common.normal, Slap_common.trans or Slap_common.conjtr).

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

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, [< `C | `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:

If trans = Slap_common.normal, then OP(a) = a;
If trans = Slap_common.trans, then OP(a) = a^T;
If trans = Slap_common.conjtr, then OP(a) = a^H (the conjugate transpose of 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, [< `C | `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:

If trans = Slap_common.normal, then OP(a) = a;
If trans = Slap_common.trans, then OP(a) = a^T;
If trans = Slap_common.conjtr, then OP(a) = a^H (the conjugate transpose of 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

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

alpha : default = 1.0

val trmv : trans:('n * 'n, 'n * 'n, [< `C | `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:

If trans = Slap_common.normal, then OP(a) = a;
If trans = Slap_common.trans, then OP(a) = a^T;
If trans = Slap_common.conjtr, then OP(a) = a^H (the conjugate transpose of a).

diag : default = Slap_common.non_unit

If diag = Slap_common.unit, then a is unit triangular;
If diag = Slap_common.non_unit, then a is not unit triangular.

up : default = Slap_common.upper

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

val trsv : trans:('n * 'n, 'n * 'n, [< `C | `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:

If trans = Slap_common.normal, then OP(a) = a;
If trans = Slap_common.trans, then OP(a) = a^T;
If trans = Slap_common.conjtr, then OP(a) = a^H (the conjugate transpose of a).

diag : default = Slap_common.non_unit

If diag = Slap_common.unit, then a is unit triangular;
If diag = Slap_common.non_unit, then a is not unit triangular.

up : default = Slap_common.upper

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

val tpmv : trans:('n * 'n, 'n * 'n, [< `C | `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:

If trans = Slap_common.normal, then OP(a) = a;
If trans = Slap_common.trans, then OP(a) = a^T;
If trans = Slap_common.conjtr, then OP(a) = a^H (the conjugate transpose of a).

diag : default = Slap_common.non_unit

If diag = Slap_common.unit, then a is unit triangular;
If diag = Slap_common.non_unit, then a is not unit triangular.

up : default = Slap_common.upper

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

val tpsv : trans:('n * 'n, 'n * 'n, [< `C | `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:

If trans = Slap_common.normal, then OP(a) = a;
If trans = Slap_common.trans, then OP(a) = a^T;
If trans = Slap_common.conjtr, then OP(a) = a^H (the conjugate transpose of a).

diag : default = Slap_common.non_unit

If diag = Slap_common.unit, then a is unit triangular;
If diag = Slap_common.non_unit, then a is not unit triangular.

up : default = Slap_common.upper

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

Level 3

val gemm : ?beta:num_type ->
       ?c:('m, 'n, 'c_cd) mat ->
       transa:('a_m * 'a_k, 'm * 'k, [< `C | `N | `T ]) trans3 ->
       ?alpha:num_type ->
       ('a_m, 'a_k, 'a_cd) mat ->
       transb:('b_k * 'b_n, 'k * 'n, [< `C | `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:

If transa = Slap_common.normal, then OP(a) = a;
If transa = Slap_common.trans, then OP(a) = a^T;
If transa = Slap_common.conjtr, then OP(a) = a^H (the conjugate transpose of a).

alpha : default = 1.0

transb : the transpose flag for b:

If transb = Slap_common.normal, then OP(b) = b;
If transb = Slap_common.trans, then OP(b) = b^T;
If transb = Slap_common.conjtr, then OP(b) = b^H (the conjugate transpose of 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

c := alpha * a * b + beta * c (if side = Slap_common.left) or
c := alpha * b * a + beta * c (if side = Slap_common.right)

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

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

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, [< `C | `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

b := alpha * OP(a) * b (if side = Slap_common.left) or
b := alpha * b * OP(a) (if side = Slap_common.right)

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

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

transa : the transpose flag for a:

If transa = Slap_common.normal, then OP(a) = a;
If transa = Slap_common.trans, then OP(a) = a^T;
If transa = Slap_common.conjtr, then OP(a) = a^H (the conjugate transpose of a).

diag : default = Slap_common.non_unit

If diag = Slap_common.unit, then a is unit triangular;
If diag = Slap_common.non_unit, then a is not unit triangular.

alpha : default = 1.0

val trsm : side:('k, 'm, 'n) Slap_common.side ->
       ?up:[< `L | `U ] Slap_common.uplo ->
       transa:('k * 'k, 'k * 'k, [< `C | `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

OP(a) * x = alpha * b (if side = Slap_common.left) or
x * OP(a) = alpha * b (if side = Slap_common.right)

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

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

transa : the transpose flag for a:

If transa = Slap_common.normal, then OP(a) = a;
If transa = Slap_common.trans, then OP(a) = a^T;
If transa = Slap_common.conjtr, then OP(a) = a^H (the conjugate transpose of a).

diag : default = Slap_common.non_unit

If diag = Slap_common.unit, then a is unit triangular;
If diag = Slap_common.non_unit, then a is not unit triangular.

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

c := alpha * a * a^T + beta * c (if trans = Slap_common.normal) or
c := alpha * a^T * a + beta * c (if trans = Slap_common.trans)

where a is a general matrix and c is a symmetric matrix.

up : default = Slap_common.upper

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

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

c := alpha * a * b^T + alpha * b * a^T + beta * c (if trans = Slap_common.normal) or
c := alpha * a^T * b + alpha * b^T * a + beta * c (if trans = Slap_common.trans)

with symmetric matrix c, and general matrices a and b.

up : default = Slap_common.upper

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

beta : default = 0.0

trans : the transpose flag for a

alpha : default = 1.0

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

If uplo = Slap_common.upper, then the upper triangular part of a is copied;
If uplo = Slap_common.lower, then the lower triangular part of a is copied.

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.

If norm = Slap_common.norm_1, the one norm is returned;
If norm = Slap_common.norm_inf, the infinity norm is returned;
If norm = Slap_common.norm_amax, the largest absolute value of elements in matrix a (not a matrix norm) is returned;
If norm = Slap_common.norm_frob, the Frobenius norm is returned.

work : default = an optimum-length vector.

val lauum : ?up:[< `L | `U ] Slap_common.uplo -> ('n, 'n, 'cd) mat -> unit

lauum ?up a computes

U * U^T where U is the upper triangular part of matrix a if up is Slap_common.upper.
L^T * L where L is the lower triangular part of matrix a if up is Slap_common.lower.

The upper or lower triangular part is overwritten.

up : default = Slap_common.upper

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

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, [< `C | `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:

If trans = Slap_common.normal, then OP(a) = a;
If trans = Slap_common.trans, then OP(a) = a^T;
If trans = Slap_common.conjtr, then OP(a) = a^H (the conjugate transpose of 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

If up = true, then the upper triangular part of a is used;
If up = false, then the lower triangular part of a is used.

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:

a = P * U * D * U^T * P^T if up = true;
a = P * L * D * L^T * P^T if up = false

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

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

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:

a = P * U * D * U^T * P^T if up = true;
a = P * L * D * L^T * P^T if up = false

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

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

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:

a = P * U * D * U^T * P^T if up = true;
a = P * L * D * L^T * P^T if up = false

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

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

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:

a = U^T * U (real) or a = U^H * U (complex) if up = true;
a = L * L^T (real) or a = L * L^H (complex) if up = false

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

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

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:

a = U^T * U (real) or a = U^H * U (complex) if up = Slap_common.upper;
a = L * L^T (real) or a = L * L^H (complex) if up = Slap_common.lower

where U and L are upper and lower triangular matrices, respectively.
Raises Failure if a is singular.

up : default = Slap_common.upper

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

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:

a = U^T * U (real) or a = U^H * U (complex) if up = Slap_common.upper;
a = L * L^T (real) or a = L * L^H (complex) if up = Slap_common.lower

where U and L are upper and lower triangular matrices, respectively.
Raises Failure if a is singular.

up : default = Slap_common.upper

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

factorize : default = true (potrf is called implicitly)

jitter : default = nothing

val trtrs : ?up:[< `L | `U ] Slap_common.uplo ->
       trans:('n * 'n, 'n * 'n, [< `C | `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

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

trans : the transpose flag for a:

If trans = Slap_common.normal, then OP(a) = a;
If trans = Slap_common.trans, then OP(a) = a^T;
If trans = Slap_common.conjtr, then OP(a) = a^H (the conjugate transpose of a).

diag : default = `N

If diag = `U, then a is unit triangular;
If diag = `N, then a is not unit triangular.

val tbtrs : kd:'kd Slap_size.t ->
       ?up:[< `L | `U ] Slap_common.uplo ->
       trans:('n * 'n, 'n * 'n, [< `C | `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

If up = Slap_common.upper, then the upper triangular part of A is used;
If up = Slap_common.lower, then the lower triangular part of A is used.

trans : the transpose flag for A:

If trans = Slap_common.normal, then OP(A) = A;
If trans = Slap_common.trans, then OP(A) = A^T;
If trans = Slap_common.conjtr, then OP(A) = A^H (the conjugate transpose of A).

diag : default = `N

If diag = `U, then A is unit triangular;
If diag = `N, then A is not unit triangular.

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

If up = Slap_common.upper, then the upper triangular part of A is used;
If up = Slap_common.lower, then the lower triangular part of A is used.

diag : default = `N

If diag = `U, then a is unit triangular;
If diag = `N, then a is not unit triangular.

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:

If up = Slap_common.upper, then a = U^T * U (real) or a = U^H * U (complex)
If up = Slap_common.lower, then a = L^T * L (real) or a = L^H * L (complex)

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

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

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.

If up = Slap_common.upper, then a = U^T * U (real) or a = U^H * U (complex)
If up = Slap_common.lower, then a = L^T * L (real) or a = L^H * L (complex)

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

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

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:

If up = Slap_common.upper, then a = U^T * U (real) or a = U^H * U (complex)
If up = Slap_common.lower, then a = L^T * L (real) or a = L^H * L (complex)

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

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

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:

If up = Slap_common.upper, then a = U * D * U^T
If up = Slap_common.lower, then a = L * D * L^T 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

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

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:

If up = Slap_common.upper, then a = U * D * U^T
If up = Slap_common.lower, then a = L * D * L^T 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

If up = Slap_common.upper, then the upper triangular part of a is used;
If up = Slap_common.lower, then the lower triangular part of a is used.

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.

If trans = Slap_common.normal and 'm >= 'n: find the least square solution to an overdetermined system by minimizing ||b - A * x||^2.
If trans = Slap_common.normal and 'm < 'n: find the minimum norm solution to an underdetermined system a * x = b.
If trans = Slap_common.trans, and 'm >= 'n: find the minimum norm solution to an underdetermined system a^H * x = b.
If trans = Slap_common.trans and 'm < 'n: find the least square solution to an overdetermined system by minimizing ||b - A^H * x||^2.

work : default = an optimum-length vector.

trans : the transpose flag for a.

BLAS interface

Level 1

val dotu : ('n, 'x_cd) vec -> ('n, 'y_cd) vec -> num_type

dotc x y computes x^T y.
Since 2.0.0
Returns an inner product of given two vectors.

val dotc : ('n, 'x_cd) vec -> ('n, 'y_cd) vec -> num_type

dotc x y computes x^H y.
Since 2.0.0
Returns an inner product of a conjugated vector with another vector.