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## sundials

### Description

The Chicken `sundials` library provides bindings to the solvers from the SUNDIALS library. SUNDIALS (SUite of Nonlinear and DIfferential/ALgebraic equation Solvers) is a collection of solvers for systems of ordinary differential equations and differential-algebraic equations.

The Chicken `sundials` library provides interfaces to the CVODE and IDA solvers and has been tested with SUNDIALS version 2.4.0.

### Library procedures

#### IDA solver interface

*[procedure]*

`(ida-create-solver TSTART TSTOP VARIABLES DERIVATIVES RESIDUAL-MAIN [RESIDUAL-INIT] [RESIDUAL-EVENT] [EVENTS] [ALG-OR-DIFF] [SUPPRESS] [IC] [USER-DATA] [RELTOL] [ABSTOL]) => IDA-SOLVER`

Creates and initializes an object representing a problem to be solved with the IDA solver.

Arguments `TSTART` and `TSTOP` must be real numbers that represent the beginning and end of the independent variable range.

Arguments `VARIABLES` and `DERIVATIVES` must be SRFI-4 `f64vector` objects that hold respectively the initial values and derivatives of the system variables.

Argument `RESIDUAL-MAIN` is used to compute the residual function `F` and must be a procedure of the following form:

(LAMBDA T YY YP DATA)

or

(LAMBDA T YY YP)

depending on whether the `USER-DATA` optional argument is set, where

`T`- real-valued independent variable
`YY`- SRFI-4
`f64vector`with current variable values `YP`- SRFI-4
`f64vector`with current variable derivatives `DATA`- is a user data object (if set)

This procedure must return a SRFI-4 `f64vector` containing the residual vector.

Optional keyword argument `RESIDUAL-EVENT` must be a procedure of the same form as `RESIDUAL-MAIN`, which computes a rootfinding problem to be solved during the integration of the system. It is set only if argument `EVENTS` is given.

Optional keyword argument `EVENTS` is an SRFI-4 `s32vector` that is used for storage of root finding solutions. It must be given if `RESIDUAL-EVENT` is given.

Optional keyword argument `ALG-OR-DIFF` must be an SRFI-4 `s32vector` which indicates the algebraic and differential variables in the system. A value of 1 indiciates differential variable, and a value of 0 indicates an algebraic one. This is required if the `SUPPRESS` argument is given and true.

Optional keyword argument `SUPPRESS` is a boolean flag that indicates whether algebraic variables must be suppressed in the local error test. If it is true (suppress), then the argument `ALG-OR-DIFF` must be given.

Optional keyword argument `IC` is a boolean flag that indicates whether the solver must calculate consistent initial conditions, or whether it must use the initial conditions given by `VARIABLES`.

Optional keyword argument `USER-DATA` is an object that will be passed as an additional argument to the residual functions.

Optional keyword arguments `RELTOL` and `ABSTOL` specify relative and absolute error tolerance, respectively. These both default to 1e-4.

*[procedure]*

`(ida-destroy-solver IDA-SOLVER)`

Deallocates the memory associated with the given solver.

*[procedure]*

`(ida-solve IDA-SOLVER T)`

Integrates the system over an interval in the independent variable. This procedure returns either when the given `T` is reached, or when a root is found.

*[procedure]*

`(ida-yy IDA-SOLVER)`

Returns the vector of current state values of the system.

*[procedure]*

`(ida-yp IDA-SOLVER)`

Returns the vector of current state derivative values of the system.

*[procedure]*

`(ida-get-last-order IDA-SOLVER)`

Returns the order used during the last solver step.

*[procedure]*

`(ida-get-last-step IDA-SOLVER)`

Returns the steps size used during the last solver step.

*[procedure]*

`(ida-get-num-steps IDA-SOLVER)`

Returns the cumulative number of steps taken by the solver.

#### CVODE solver interface

<procedure>(cvode-create-solver TSTART TSTOP VARIABLES RHS-FN

[LMM] [ITER] [EWT-FN] [EVENT-FN] [EVENTS] [USER-DATA] [RELTOL] [ABSTOL]) => CVODE-SOLVER</procedure>

Creates and initializes an object representing a problem to be solved with the CVODE solver.

Arguments `TSTART` and `TSTOP` must be real numbers that represent the beginning and end of the independent variable range.

Arguments `VARIABLES` must be a SRFI-4 `f64vector` object that holds the initial values of the system variables.

Argument `RHS-FN` is used to compute the right-hand side of the equations, and must be a procedure of the following form:

(LAMBDA T YY DATA)

or

(LAMBDA T YY)

depending on whether the `USER-DATA` optional argument is set, where

`T`- real-valued independent variable
`YY`- SRFI-4
`f64vector`with current variable values `DATA`- is a user data object (if set)

This procedure must return a SRFI-4 `f64vector` containing the residual vector.

Optional keyword argument `EWT-FN` must be a procedure of the same form as `(LAMBDA YY)`, which computes error weights for the system variables, and which can be used in place of relative and absolute error tolerance.

Optional keyword argument `EVENT-FN` must be a procedure of the same form as `RHS-FN`, which computes a rootfinding problem to be solved during the integration of the system. It is set only if argument `EVENTS` is given.

Optional keyword argument `EVENTS` is an SRFI-4 `s32vector` that is used for storage of root finding solutions. It must be given if `EVENT-FN` is given.

Optional keyword argument `LMM` specifies the linear multistep method to be used and can be one of `cvode-lmm/adams` (default) or `cvode-lmm/bdf`. `cvode-lmm/bdf` is recommended for stiff problems.

Optional keyword argument `ITER` specifies the iteration type to be used and can be one of `cvode-iter/functional` (default) or `cvode-iter/newton`. `cvode-iter/newton` is recommended for stiff problems.

Optional keyword argument `USER-DATA` is an object that will be passed as an additional argument to the residual functions.

Optional keyword arguments `RELTOL` and `ABSTOL` specify relative and absolute error tolerance, respectively. These both default to 1e-4. They are only set of `EWT-FN` is not specified.

### Example

;; ;; Hodgkin-Huxley model ;; (use mathh sundials srfi-4) (define neg -) (define pow expt) (define TEND 500.0) ;; Model parameters (define (I_stim t) 10) (define C_m 1) (define E_Na 50) (define E_K -77) (define E_L -54.4) (define gbar_Na 120) (define gbar_K 36) (define g_L 0.3) ;; Rate functions (define (amf v) (* 0.1 (/ (+ v 40) (- 1.0 (exp (/ (neg (+ v 40)) 10)))))) (define (bmf v) (* 4.0 (exp (/ (neg (+ v 65)) 18)))) (define (ahf v) (* 0.07 (exp (/ (neg (+ v 65)) 20)))) (define (bhf v) (/ 1.0 (+ 1.0 (exp (/ (neg (+ v 35)) 10))))) (define (anf v) (* 0.01 (/ (+ v 55) (- 1 (exp (/ (neg (+ v 55)) 10)))))) (define (bnf v) (* 0.125 (exp (/ (neg (+ v 65)) 80)))) ;; State functions (define (minf v) (* 0.5 (+ 1 (tanh (/ (- v v1) v2))))) (define (winf v) (* 0.5 (+ 1 (tanh (/ (- v v3) v4))))) (define (lamw v) (* phi (cosh (/ (- v v3) (* 2 v4))))) ;; Model equations (define (rhs t yy) (let ((v (f64vector-ref yy 0)) (m (f64vector-ref yy 1)) (h (f64vector-ref yy 2)) (n (f64vector-ref yy 3))) ;; transition rates at current step (let ((am (amf v)) (an (anf v)) (ah (ahf v)) (bm (bmf v)) (bn (bnf v)) (bh (bhf v)) (g_Na (* gbar_Na (* h (pow m 3)))) (g_K (* gbar_K (pow n 4)))) (let ( ;; currents (I_Na (* (- v E_Na) g_Na)) (I_K (* (- v E_K) g_K)) (I_L (* g_L (- v E_L)))) (let ( ;; state equations (dm (- (* am (- 1 m)) (* bm m))) (dh (- (* ah (- 1 h)) (* bh h))) (dn (- (* an (- 1 n)) (* bn n))) (dv (/ (- (I_stim t) I_L I_Na I_K) C_m)) ) (f64vector dv dm dh dn) ))) )) (let ((yy (f64vector -65 0.052 0.596 0.317)) ;; v m h n ;; Integration limits (t0 0.0) (tf TEND) (dt 1e-2)) ;; CVODE initialization (let ((solver (cvode-create-solver t0 tf yy rhs abstol: 1e-4 reltol: 1e-4))) ;; In loop, call CVodeSolve, print results, and test for error. (let recur ((tnext (+ t0 dt)) (iout 1)) (let ((flag (cvode-solve solver tnext))) (if (negative? flag) (error 'main "CVODE solver error" flag)) (print-results solver tnext) (if (< tnext tf) (recur (+ tnext dt) (+ 1 iout))) )) (cvode-destroy-solver solver) (define (print-results solver t) (let ((yy (cvode-yy solver))) (printf "~A ~A ~A ~A ~A ~A ~A ~A~%" t (f64vector-ref yy 0) (f64vector-ref yy 1) (f64vector-ref yy 2) (f64vector-ref yy 3) (cvode-get-last-order solver) (cvode-get-num-steps solver) (cvode-get-last-step solver) )))

### Version History

- 1.0 Initial release

### License

Copyright 2011 Ivan Raikov. All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. Neither the name of the author nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDERS OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.